Introduction
The relationship between Floer-theoretic knot invariants of Ozsváth and Szabó [12 ] and of Rasmussen [16 ] and quantum knot invariants of Khovanov [8 ] and of Khovanov and Rozansky [9 , 10 ] has been an active topic in topology for the past twenty-five years. These invariants categorify many knot invariants. That is, the graded Euler characteristic of the knot Floer homology of a knot returns its Alexander polynomial, and the graded Euler characteristic of the Khovanov homology of a knot returns its Jones polynomial. The first relationship between them came from a spectral sequence from the Khovanov homology of a link to the Heegaard Floer homology of its double branched cover, constructed by Ozsváth and Szabó in [14 ] . Since then, a slew of spectral sequences of that form have been discovered, yielding a variety of rank inequalities between the associated homologies as well as indicating the possibility of more interesting relationships.
Consequently, the following was a long-standing conjecture of Rasmussen [17 ] .
Theorem 1.1 (Rasmussen’s Conjecture). [2 ] For a knot K ⊂ S 3 K \subset S^3 K ⊂ S 3 rk ( K h ~ ( K ; Q ) ) ≥ rk ( H F K ^ ( K ; Q ) ) \operatorname{rk}(\widetilde{Kh}(K;\mathbb{Q})) \ge\operatorname{rk}(\widehat{HFK}(K;\mathbb{Q})) rk ( K h ( K ; Q )) ≥ rk ( H F K ( K ; Q )) K h ~ \widetilde{Kh} K h
In [17 ] , Rasmussen suggests that constructing a spectral sequence is a promising way to address this conjecture, a strategy successfully employed by Dowlin in [1 , 2 ] . It is natural to want to complete the diagram in Figure 1 with spectral sequences from the Khovanov-Rozansky homologies, K R n ( K ) KR_n(K) K R n ( K ) [1 ] , Dowlin defines a candidate family of link invariants, called H F K n ( K ) HFK_n(K) H F K n ( K )
Figure 1. Dotted arrows are conjectured, while solid arrows have been proven in suitable versions: the curved arrows on the left are Rasmussen’s spectral sequences [18 ] , those on the right are due to Dowlin [1 Lemma 2.23] , and the bottom arrow is Dowlin’s spectral sequence from reduced Khovanov homology to H F K ^ \widehat{HFK} H F K [2 ] .
Conjecture 1.2. [[1 Conjecture 1.6] ] For a link L ⊂ S 3 L \subset S^3 L ⊂ S 3 K R n ( L ) ⇒ H F K n ( L ) KR_n(L) \Rightarrow HFK_n(L) K R n ( L ) ⇒ H F K n ( L ) K R ‾ n ( L ) ⇒ H F K ^ n ( L ) \overline{KR}_n(L) \Rightarrow\widehat{HFK}_n(L) K R n ( L ) ⇒ H F K n ( L ) [1 Conjecture 1.6] ].
Motivated by Conjecture 1.2, we study H F K n ( L ) HFK_n(L) H F K n ( L ) L L L
Conjecture 1.3. For any knot K K K S 3 S^3 S 3 N ( K ) N(K) N ( K ) n ≥ N ( K ) n \ge N(K) n ≥ N ( K ) H F K n ( K ) HFK_n(K) H F K n ( K ) H F K − ( K ) HFK^{-}(K) H F K − ( K )
H F K n ( K ) ≅ ( H F K − ( K ) ⊗ Q [ U ] Q [ U ] ( U n ) ) ⊕ Tor 1 Q [ U ] ( H F K − ( K ) ; Q [ U ] ( U n ) ) . HFK_n(K) \cong\left(HFK^{-}(K) \otimes_{\mathbb{Q}[U]} \frac{\mathbb{Q}[U]}{(U^n)}\right) \oplus\operatorname{Tor}^{\mathbb{Q}[U]}_1\left(HFK^{-}(K); \frac{\mathbb{Q}[U]}{(U^n)}\right). H F K n ( K ) ≅ ( H F K − ( K ) ⊗ Q [ U ] ( U n ) Q [ U ] ) ⊕ Tor 1 Q [ U ] ( H F K − ( K ) ; ( U n ) Q [ U ] ) . Although further investigation is required to prove Conjecture 1.3, much can be said about two wide classes of knots. Section 3 presents supporting examples, including the trefoil (an L L L T 3 , 4 T_{3,4} T 3 , 4 Y Y Y L L L rk H F ^ ( Y ; Q ) = ∣ H 1 ( Y ; Z ) ∣ \operatorname{rk}\widehat{HF}(Y;\mathbb{Q})=|H_1(Y;\mathbb{Z})| rk H F ( Y ; Q ) = ∣ H 1 ( Y ; Z ) ∣ 1 L L L L L L [13 ] ], the Alexander polynomial of an L L L Δ K ( t ) = ∑ j = 0 m − 1 ( − 1 ) j t e j \Delta_K(t)=\sum_{j=0}^{m-1}(-1)^j t^{e_j} Δ K ( t ) = ∑ j = 0 m − 1 ( − 1 ) j t e j e 0 > e 1 > ⋯ > e m − 1 e_0>e_1>\cdots>e_{m-1} e 0 > e 1 > ⋯ > e m − 1 e j − 1 − e j e_{j-1}-e_j e j − 1 − e j Δ K \Delta_K Δ K K K K H F K ^ ( K ; Q ) \widehat{HFK}(K;\mathbb{Q}) H F K ( K ; Q ) δ \delta δ
Theorem 1.4. Let n ≥ 2 n \ge2 n ≥ 2 K ⊂ S 3 K \subset S^3 K ⊂ S 3 L L L Δ K \Delta_K Δ K n − 1 n-1 n − 1 Q [ U ] \mathbb{Q}[U] Q [ U ]
H F K n ( K ) ≅ ( H F K − ( K ) ⊗ Q [ U ] Q [ U ] ( U n ) ) ⊕ Tor 1 Q [ U ] ( H F K − ( K ) ; Q [ U ] ( U n ) ) . HFK_n(K) \cong\left(HFK^{-}(K) \otimes_{\mathbb{Q}[U]} \frac{\mathbb{Q}[U]}{(U^n)}\right) \oplus\operatorname{Tor}^{\mathbb{Q}[U]}_1\left(HFK^{-}(K); \frac{\mathbb{Q}[U]}{(U^n)}\right). H F K n ( K ) ≅ ( H F K − ( K ) ⊗ Q [ U ] ( U n ) Q [ U ] ) ⊕ Tor 1 Q [ U ] ( H F K − ( K ) ; ( U n ) Q [ U ] ) . That is to say, for such knots, H F K n ( K ) HFK_n(K) H F K n ( K ) H F K − ( K ) HFK^{-}(K) H F K − ( K )
Corollary 1.5. For any L L L K K K H F K n ( K ) HFK_n(K) H F K n ( K ) K K K H F K n ( K ) HFK_n(K) H F K n ( K ) τ ( K ) \tau(K) τ ( K ) n ≥ 2 n \ge2 n ≥ 2
H F K n ( K ) ≅ Q [ U ] ( U n ) ⊕ Q det ( K ) − 1 HFK_n(K) \cong\frac{\mathbb{Q}[U]}{(U^n)} \oplus\mathbb{Q}^{\det(K)-1} H F K n ( K ) ≅ ( U n ) Q [ U ] ⊕ Q d e t ( K ) − 1 as ungraded Q [ U ] \mathbb{Q}[U] Q [ U ]
While the first part of Corollary 1.5 is implicitly known from the definition of H F K n ( L ) HFK_n(L) H F K n ( L ) [13 ] ], where we see that the knot Floer complex (and hence H F K − ( K ) HFK^{-}(K) H F K − ( K ) L L L [15 ] ], where we see that the knot Floer complex of any thin knot is determined from its Alexander polynomial and τ \tau τ τ = − σ / 2 \tau=-\sigma/2 τ = − σ /2 [11 ] ], so for these knots the signature may be used in place of τ \tau τ [15 ] ] will play a more critical role in the proof for thin knots in Section 4.
The restriction to large n n n n = 1 n=1 n = 1 H F K 1 ( K ) ≅ H F ^ ( S 3 ) ≅ Q HFK_1(K)\cong\widehat{HF}(S^3)\cong\mathbb{Q} H F K 1 ( K ) ≅ H F ( S 3 ) ≅ Q H F K ^ ( K ) \widehat{HFK}(K) H F K ( K ) L L L H F K n ( K ) HFK_n(K) H F K n ( K ) n ≥ 2 n\ge2 n ≥ 2 Δ K \Delta_K Δ K
Corollary 1.6. Let 2 ≤ p < q 2\le p<q 2 ≤ p < q n ≥ 2 n\ge2 n ≥ 2 T p , q T_{p,q} T p , q n ≥ p n\ge p n ≥ p T 3 , 4 T_{3,4} T 3 , 4 n = 2 n=2 n = 2 dim Q H F K 2 ( T 3 , 4 ) = 6 \dim_{\mathbb{Q}}HFK_2(T_{3,4})=6 dim Q H F K 2 ( T 3 , 4 ) = 6 8 8 8
Organization
In Section 2, we introduce and review a construction of the invariants H F K n ( K ) HFK_n(K) H F K n ( K ) [1 ] ]. Section 3 compares several computations of these invariants against older knot Floer homology flavors for representatives from various classes of knots. In Section 4, we prove the main theorem.
All results of this note are due to the author. Claude Opus 5.5 was used in proofreading, checking proof details, and generating and checking examples.
Acknowledgments
This note is modified from the author’s Ph.D. dissertation at Michigan State University. I am eternally grateful to my Ph.D. advisor Matthew Hedden for helpful conversation and comments on early drafts. I thank Charles Stine and Shea Vela-Vick for helpful conversations. I also thank Chris St. Clair, Dean Spyropoulos, and Chen Zhang for comments on early drafts.
I also want to thank the National Science Foundation for support with grants DMS-2104664, the RTG NSF Award DMS-2135960, DMS-1709016, and the RTG NSF Award DMS-2231492. Their financial support made this research possible.
Background
Heegaard Diagrams
A common way to depict 3-manifolds in low-dimensional topology is using a Heegaard diagram, which presents a convenient Morse decomposition of the 3-manifold. These can be enhanced to encode embedded links in the 3-manifold as well.
Definition 2.1. (cf. [1 Definition 2.1] ) A (multi-pointed) Heegaard diagram H \mathcal{H} H L L L Y Y Y ( Σ , α , β , w , z ) (\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{w},\mathbf{z}) ( Σ , α , β , w , z )
Σ \Sigma Σ g g g
α \boldsymbol{\alpha} α β \boldsymbol{\beta} β g + k − 1 g+k-1 g + k − 1 { α 1 , … , α g + k − 1 } \{\alpha_1,\ldots,\alpha_{g+k-1}\} { α 1 , … , α g + k − 1 } { β 1 , … , β g + k − 1 } \{\beta_1,\ldots,\beta_{g+k-1}\} { β 1 , … , β g + k − 1 } Σ \Sigma Σ
α \boldsymbol{\alpha} α β \boldsymbol{\beta} β g g g H 1 ( Σ ; Z ) H_1(\Sigma;\mathbb{Z}) H 1 ( Σ ; Z )
w \mathbf{w} w z \mathbf{z} z k k k { w 1 , … , w k } \{w_1,\ldots,w_k\} { w 1 , … , w k } { z 1 , … , z k } \{z_1,\ldots,z_k\} { z 1 , … , z k } k k k L L L Σ ∖ α \Sigma\setminus\boldsymbol{\alpha} Σ ∖ α Σ ∖ β \Sigma\setminus\boldsymbol{\beta} Σ ∖ β w w w z z z
The manifold Y Y Y Σ \Sigma Σ Σ × [ 0 , 1 ] \Sigma\times[0,1] Σ × [ 0 , 1 ] Σ × { 0 } \Sigma\times\{0\} Σ × { 0 } Σ × { 1 } \Sigma\times\{1\} Σ × { 1 }
The link L L L Y Y Y Σ \Sigma Σ w \mathbf{w} w z \mathbf{z} z w ⊔ z × [ 0 , 1 ] ⊂ Σ × [ 0 , 1 ] \mathbf{w}\sqcup\mathbf{z}\times[0,1]\subset\Sigma\times[0,1] w ⊔ z × [ 0 , 1 ] ⊂ Σ × [ 0 , 1 ] Σ ∖ α \Sigma\setminus\boldsymbol{\alpha} Σ ∖ α w w w z z z Σ × { 0 } \Sigma\times\{0\} Σ × { 0 } S 2 S^2 S 2 Σ × { 1 } \Sigma\times\{1\} Σ × { 1 }
Definition 2.2. In [1 Definition 2.2] , Dowlin defined a notion of a punctured Heegaard diagram , which adds to ( Σ , α , β , w , z ) (\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{w},\mathbf{z}) ( Σ , α , β , w , z )
A basepoint p p p puncture .
A curve α g + k \alpha_{g+k} α g + k β g + k \beta_{g+k} β g + k p p p w \mathbf{w} w z \mathbf{z} z α \boldsymbol{\alpha} α β \boldsymbol{\beta} β g g g H 1 ( Σ ; Z ) H_1(\Sigma;\mathbb{Z}) H 1 ( Σ ; Z )
Remark 2.3. In the above construction of the pair ( Y , L ) (Y,L) ( Y , L ) L ⊔ U L\sqcup\mathcal{U} L ⊔ U U \mathcal{U} U w 0 w_0 w 0 z 0 z_0 z 0 p p p [1 Remark 2.3] .
Knot Floer Homology
Here, we provide the groundwork for Dowlin’s new family of invariants in [1 ] . For another overview of knot Floer homology, we direct the reader to [6 ] . For greater detail, see [12 ] .
All of our definitions will be made over Q \mathbb{Q} Q [1 , 2 ] , the tools needed depend on working with a field of characteristic zero (we need n n n [2 ] , to be invertible), at the cost of keeping track of signs. However, most definitions here can be adapted to one’s preferred field for other applications. Knot Floer complexes come in a variety of flavors, but we are only interested in two: the minus flavor C F K − ( L ) CFK^{-}(L) C F K − ( L ) C F K U , V ( L ) CFK_{U,V}(L) C F K U , V ( L )
Given a multi-pointed unpunctured Heegaard diagram H \mathcal{H} H L ⊂ Y L\subset Y L ⊂ Y master complex C F K U , V ( H ) CFK_{U,V}(\mathcal{H}) C F K U , V ( H ) Q [ U 1 , … , U k , V 1 , … , V k ] \mathbb{Q}[U_1,\ldots,U_k,V_1,\ldots,V_k] Q [ U 1 , … , U k , V 1 , … , V k ] T α ∩ T β \mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta} T α ∩ T β T α \mathbb{T}_{\alpha} T α T β \mathbb{T}_{\beta} T β Sym g + k − 1 ( Σ ) \operatorname{Sym}^{g+k-1}(\Sigma) Sym g + k − 1 ( Σ ) Sym g + k ( Σ ) \operatorname{Sym}^{g+k}(\Sigma) Sym g + k ( Σ ) α \alpha α β \beta β T β \mathbb{T}_{\beta} T β H \mathcal{H} H x x x y y y
∂ U , V ( x ) = ∑ y ∈ T α ∩ T β ∑ ϕ ∈ π 2 ( x , y ) μ ( ϕ ) = 1 # M ^ ( ϕ ) U 1 n w 1 ( ϕ ) ⋯ U k n w k ( ϕ ) V 1 n z 1 ( ϕ ) ⋯ V k n z k ( ϕ ) y . \partial_{U,V}(x)=\sum_{y\in\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}}\sum_{\substack{\phi\in\pi_2(x,y)\\ \mu(\phi)=1}}\#\widehat{\mathcal{M}}(\phi)U_1^{n_{w_1}(\phi)}\cdots U_k^{n_{w_k}(\phi)}V_1^{n_{z_1}(\phi)}\cdots V_k^{n_{z_k}(\phi)}y. ∂ U , V ( x ) = y ∈ T α ∩ T β ∑ ϕ ∈ π 2 ( x , y ) μ ( ϕ ) = 1 ∑ # M ( ϕ ) U 1 n w 1 ( ϕ ) ⋯ U k n w k ( ϕ ) V 1 n z 1 ( ϕ ) ⋯ V k n z k ( ϕ ) y . Remark 2.4. [1 Lemma 2.4] In general, C F K U , V ( H ) CFK_{U,V}(\mathcal{H}) C F K U , V ( H ) w a ( i ) w_{a(i)} w a ( i ) z i z_i z i w b ( i ) w_{b(i)} w b ( i ) z i z_i z i
∂ U , V 2 = ∑ i = 1 k ( U a ( i ) − U b ( i ) ) V i \partial_{U,V}^{2}=\sum_{i=1}^{k}(U_{a(i)}-U_{b(i)})V_i ∂ U , V 2 = i = 1 ∑ k ( U a ( i ) − U b ( i ) ) V i When each component of L L L C F K U , V ( H ) CFK_{U,V}(\mathcal{H}) C F K U , V ( H )
The complex C F K − ( H ) CFK^{-}(\mathcal{H}) C F K − ( H ) Q [ U 1 , … , U k ] \mathbb{Q}[U_1,\ldots,U_k] Q [ U 1 , … , U k ] T α ∩ T β \mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta} T α ∩ T β
∂ − ( x ) = ∑ y ∈ T α ∩ T β ∑ ϕ ∈ π 2 ( x , y ) μ ( ϕ ) = 1 n z ( ϕ ) = 0 # M ^ ( ϕ ) U 1 n w 1 ( ϕ ) ⋯ U k n w k ( ϕ ) y \partial^{-}(x)=\sum_{y\in\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}}\ \sum_{\substack{\phi\in\pi_2(x,y)\\ \mu(\phi)=1\\ n_z(\phi)=0}}\#\widehat{\mathcal{M}}(\phi)U_1^{n_{w_1}(\phi)}\cdots U_k^{n_{w_k}(\phi)}y ∂ − ( x ) = y ∈ T α ∩ T β ∑ ϕ ∈ π 2 ( x , y ) μ ( ϕ ) = 1 n z ( ϕ ) = 0 ∑ # M ( ϕ ) U 1 n w 1 ( ϕ ) ⋯ U k n w k ( ϕ ) y The definition of C F K U , V ( L ) CFK_{U,V}(L) C F K U , V ( L ) p p p
∂ U , V ( x ) = ∑ y ∈ T α ∩ T β ∑ ϕ ∈ π 2 ( x , y ) μ ( ϕ ) = 1 n p ( ϕ ) = 0 # M ^ ( ϕ ) U 1 n w 1 ( ϕ ) ⋯ U k n w k ( ϕ ) V 1 n z 1 ( ϕ ) ⋯ V k n z k ( ϕ ) y . \partial_{U,V}(x)=\sum_{y\in\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}}\ \sum_{\substack{\phi\in\pi_2(x,y)\\ \mu(\phi)=1\\ n_p(\phi)=0}}\#\widehat{\mathcal{M}}(\phi)U_1^{n_{w_1}(\phi)}\cdots U_k^{n_{w_k}(\phi)}V_1^{n_{z_1}(\phi)}\cdots V_k^{n_{z_k}(\phi)}y. ∂ U , V ( x ) = y ∈ T α ∩ T β ∑ ϕ ∈ π 2 ( x , y ) μ ( ϕ ) = 1 n p ( ϕ ) = 0 ∑ # M ( ϕ ) U 1 n w 1 ( ϕ ) ⋯ U k n w k ( ϕ ) V 1 n z 1 ( ϕ ) ⋯ V k n z k ( ϕ ) y . The chain homotopy type of each complex is an invariant of the link L L L C F K U , V CFK_{U,V} C F K U , V C F K U , V ( L ) CFK_{U,V}(L) C F K U , V ( L ) C F K − ( L ) CFK^{-}(L) C F K − ( L ) C F K − ( L ) CFK^{-}(L) C F K − ( L ) H F K − ( L ) = H ∗ ( C F K − ( L ) ) HFK^{-}(L)=H_{*}(CFK^{-}(L)) H F K − ( L ) = H ∗ ( C F K − ( L ))
Remark 2.5. The thin knots considered in this note enjoy another noteworthy trait: C F K U , V ( K ) CFK_{U,V}(K) C F K U , V ( K ) U U U V V V [15 ] and presented quickly in § 4 (Theorem 4.9 ). This fails for L L L C F K U , V ( T 3 , 4 ) CFK_{U,V}(T_{3,4}) C F K U , V ( T 3 , 4 )
Remark 2.6. Since, in our setting, the manifold Y Y Y S 3 S^3 S 3 Spin c \operatorname{Spin}^{c} Spin c
The Family H F K n HFK_n H F K n
In [1 ] , Dowlin constructs a family of link invariants designed to mimic features of the s l n \mathfrak{sl}_n sl n [9 ] .
Definition 2.7. [1 Definition 2.5] Given a multi-pointed punctured Heegaard diagram H \mathcal{H} H L L L Y Y Y C F K n ( H ) CFK_n(\mathcal{H}) C F K n ( H )
C F K n ( H ) = C F K U , V ( H ) / ( V i − U a ( i ) n − U b ( i ) n U a ( i ) − U b ( i ) ) CFK_n(\mathcal{H})=CFK_{U,V}(\mathcal{H})\mathbin{/}\left(V_i-\frac{U_{a(i)}^n-U_{b(i)}^n}{U_{a(i)}-U_{b(i)}}\right) C F K n ( H ) = C F K U , V ( H ) / ( V i − U a ( i ) − U b ( i ) U a ( i ) n − U b ( i ) n ) where a ( i ) a(i) a ( i ) b ( i ) b(i) b ( i ) ∑ j = 0 n − 1 U a ( i ) j U b ( i ) n − 1 − j \sum_{j=0}^{n-1}U_{a(i)}^jU_{b(i)}^{n-1-j} ∑ j = 0 n − 1 U a ( i ) j U b ( i ) n − 1 − j ∂ n \partial_n ∂ n ∂ U , V \partial_{U,V} ∂ U , V
Remark 2.8. In the case where L L L a ( 1 ) = b ( 1 ) a(1)=b(1) a ( 1 ) = b ( 1 ) ∑ j = 0 n − 1 U j U n − 1 − j = n U n − 1 \sum_{j=0}^{n-1}U^jU^{n-1-j}=nU^{n-1} ∑ j = 0 n − 1 U j U n − 1 − j = n U n − 1
V = d d U ( U n ) = n U n − 1 . V=\frac{d}{dU}(U^n)=nU^{n-1}. V = d U d ( U n ) = n U n − 1 . Theorem 2.9. [1 Lemma 2.6] The map ∂ n : C F K n ( H ) → C F K n ( H ) \partial_n:CFK_n(\mathcal{H})\to CFK_n(\mathcal{H}) ∂ n : C F K n ( H ) → C F K n ( H ) ∂ n 2 = 0 \partial_n^2=0 ∂ n 2 = 0
Let H F K n ( H ) = H ∗ ( C F K n ( H ) , ∂ n ) HFK_n(\mathcal{H})=H_*(CFK_n(\mathcal{H}),\partial_n) H F K n ( H ) = H ∗ ( C F K n ( H ) , ∂ n )
Theorem 2.10. [1 Theorem 2.7] If H 1 \mathcal{H}_1 H 1 H 2 \mathcal{H}_2 H 2 L L L ℓ \ell ℓ C F K n ( H 1 ) CFK_n(\mathcal{H}_1) C F K n ( H 1 ) C F K n ( H 2 ) CFK_n(\mathcal{H}_2) C F K n ( H 2 ) Q [ U 1 , … , U ℓ ] \mathbb{Q}[U_1,\ldots,U_\ell] Q [ U 1 , … , U ℓ ]
Definition 2.11. [[1 ] , Definition 2.8] Define H F K n ( L ) = H ∗ ( C F K n ( H ) ) HF K_n(L)=H_\ast(CFK_n(\mathcal{H})) H F K n ( L ) = H ∗ ( C F K n ( H )) H \mathcal{H} H L L L
We perform some computations in § 3 . These examples illustrate that, for knots, one has a simpler equivalent definition of H F K n ( K ) HF K_n(K) H F K n ( K ) L = K L=K L = K H F K n ( K ) HF K_n(K) H F K n ( K ) C F K U , V ( K ) CFK_{U,V}(K) C F K U , V ( K )
Theorem 2.12. [[1 ] , Lemma 2.16] If H \mathcal{H} H K K K S 3 S^3 S 3 w w w z z z
H F K n ( K ) = H ∗ ( C F K U , V ( H ) ( U V , V − n U n − 1 ) ) . HF K_n(K)=H_\ast\left(\frac{CFK_{U,V}(\mathcal{H})}{(UV,V-nU^{n-1})}\right). H F K n ( K ) = H ∗ ( ( U V , V − n U n − 1 ) C F K U , V ( H ) ) . Computational Evidence
The master complex C F K U , V ( L ) CFK_{U,V}(L) C F K U , V ( L ) [13 ] ]. An application of Theorem 2.12 simplifies the computations.
Example 3.2. The Trefoil.
As mentioned above, C F K U , V ( T p , q ) CFK_{U,V}(T_{p,q}) C F K U , V ( T p , q ) Δ ( T 2 , 3 ) = t − 1 − 1 + t \Delta(T_{2,3})=t^{-1}-1+t Δ ( T 2 , 3 ) = t − 1 − 1 + t t t t U U U V V V
The complex C F K n ( T 2 , 3 ) CFK_n(T_{2,3}) C F K n ( T 2 , 3 ) C F K − ( K ) CFK^-(K) C F K − ( K ) C F K n ( K ) CFK_n(K) C F K n ( K ) Figure 2 . On the level of homology, for n ≥ 2 n\geq2 n ≥ 2
H F K n ( T 2 , 3 ) ≅ Q [ U ] ( U n ) ⊕ Q ⊕ Q HF K_n(T_{2,3})\cong\frac{\mathbb{Q}[U]}{(U^n)}\oplus\mathbb{Q}\oplus\mathbb{Q} H F K n ( T 2 , 3 ) ≅ ( U n ) Q [ U ] ⊕ Q ⊕ Q H F K − ( T 2 , 3 ) ≅ Q [ U ] ⊕ Q . HFK^-(T_{2,3})\cong\mathbb{Q}[U]\oplus\mathbb{Q}. H F K − ( T 2 , 3 ) ≅ Q [ U ] ⊕ Q . :::`
Figure 2. On the left: the complex C F K − ( T 2 , 3 ) CFK^-(T_{2,3}) C F K − ( T 2 , 3 ) C F K n ( T 2 , 3 ) CFK_n(T_{2,3}) C F K n ( T 2 , 3 )
Notice that since Tor 1 Q [ U ] ( Q ⊕ Q [ U ] , Q [ U ] ( U n ) ) ≅ Q \operatorname{Tor}_{1}^{\mathbb{Q}[U]}\left(\mathbb{Q}\oplus\mathbb{Q}[U],\frac{\mathbb{Q}[U]}{(U^{n})}\right)\cong\mathbb{Q} Tor 1 Q [ U ] ( Q ⊕ Q [ U ] , ( U n ) Q [ U ] ) ≅ Q
H F K n ( T 2 , 3 ) ≅ H F K − ( T 2 , 3 ) ⊗ Q [ U ] Q [ U ] ( U n ) ⊕ Tor 1 Q [ U ] ( H F K − ( T 2 , 3 ) ; Q [ U ] ( U n ) ) HFK_{n}(T_{2,3})\cong HFK^{-}(T_{2,3})\otimes_{\mathbb{Q}[U]}\frac{\mathbb{Q}[U]}{(U^{n})}\oplus\operatorname{Tor}_{1}^{\mathbb{Q}[U]}\left(HFK^{-}(T_{2,3});\frac{\mathbb{Q}[U]}{(U^{n})}\right) H F K n ( T 2 , 3 ) ≅ H F K − ( T 2 , 3 ) ⊗ Q [ U ] ( U n ) Q [ U ] ⊕ Tor 1 Q [ U ] ( H F K − ( T 2 , 3 ) ; ( U n ) Q [ U ] ) ≅ ( Q [ U ] ⊕ Q ) ⊗ Q [ U ] Q [ U ] ( U n ) ⊕ Q \cong(\mathbb{Q}[U]\oplus\mathbb{Q})\otimes_{\mathbb{Q}[U]}\frac{\mathbb{Q}[U]}{(U^{n})}\oplus\mathbb{Q} ≅ ( Q [ U ] ⊕ Q ) ⊗ Q [ U ] ( U n ) Q [ U ] ⊕ Q ≅ Q [ U ] ( U n ) ⊕ Q ⊕ Q . \cong\frac{\mathbb{Q}[U]}{(U^{n})}\oplus\mathbb{Q}\oplus\mathbb{Q}. ≅ ( U n ) Q [ U ] ⊕ Q ⊕ Q . (For n = 1 n=1 n = 1 H F K 1 ( T 2 , 3 ) ≅ Q HFK_{1}(T_{2,3})\cong\mathbb{Q} H F K 1 ( T 2 , 3 ) ≅ Q Remark 2.13 .) We see similar behavior for a knot whose master complex has boxes and is not a torus knot.
Example 3.3. The Figure-Eight.
We first look at the master complex for 4 1 4_{1} 4 1 Figure 3 . The invariants for 4 1 4_{1} 4 1 n ≥ 2 n\geq2 n ≥ 2
Figure 3. On the top: the complex C F K U , V ( 4 1 ) CFK_{U,V}(4_{1}) C F K U , V ( 4 1 ) Q [ U , V ] \mathbb{Q}[U,V] Q [ U , V ] C F K − ( 4 1 ) CFK^{-}(4_{1}) C F K − ( 4 1 ) Q [ U ] \mathbb{Q}[U] Q [ U ] C F K n ( 4 1 ) CFK_{n}(4_{1}) C F K n ( 4 1 ) Q [ U ] ( U n ) \frac{\mathbb{Q}[U]}{(U^{n})} ( U n ) Q [ U ]
H F K n ( 4 1 ) ≅ Q ⊕ Q ⊕ Q ⊕ Q ⊕ Q [ U ] ( U n ) HFK_{n}(4_{1})\cong\mathbb{Q}\oplus\mathbb{Q}\oplus\mathbb{Q}\oplus\mathbb{Q}\oplus\frac{\mathbb{Q}[U]}{(U^{n})} H F K n ( 4 1 ) ≅ Q ⊕ Q ⊕ Q ⊕ Q ⊕ ( U n ) Q [ U ] H F K − ( 4 1 ) ≅ Q [ U ] ⊕ Q ⊕ Q . HFK^{-}(4_{1})\cong\mathbb{Q}[U]\oplus\mathbb{Q}\oplus\mathbb{Q}. H F K − ( 4 1 ) ≅ Q [ U ] ⊕ Q ⊕ Q . Again, the universal coefficient theorem gives us the following:
H F K n ( 4 1 ) ≅ H F K − ( 4 1 ) ⊗ Q [ U ] Q [ U ] ( U n ) ⊕ Tor 1 Q [ U ] ( H F K − ( 4 1 ) ; Q [ U ] ( U n ) ) HFK_{n}(4_{1})\cong HFK^{-}(4_{1})\otimes_{\mathbb{Q}[U]}\frac{\mathbb{Q}[U]}{(U^{n})}\oplus\operatorname{Tor}_{1}^{\mathbb{Q}[U]}\left(HFK^{-}(4_{1});\frac{\mathbb{Q}[U]}{(U^{n})}\right) H F K n ( 4 1 ) ≅ H F K − ( 4 1 ) ⊗ Q [ U ] ( U n ) Q [ U ] ⊕ Tor 1 Q [ U ] ( H F K − ( 4 1 ) ; ( U n ) Q [ U ] ) ≅ Q [ U ] ( U n ) ⊕ Q ⊕ Q ⊕ Q ⊕ Q ⊕ Q . \cong\frac{\mathbb{Q}[U]}{(U^{n})}\oplus\mathbb{Q}\oplus\mathbb{Q}\oplus\mathbb{Q}\oplus\mathbb{Q}\oplus\mathbb{Q}. ≅ ( U n ) Q [ U ] ⊕ Q ⊕ Q ⊕ Q ⊕ Q ⊕ Q . Remark 3.4. While one might wish to explore examples with diagonal arrows, we note that all diagonal arrows in the master complex for a knot disappear when descending to any of the complexes at hand.
Example 3.5. The Torus Knot T 3 , 4 T_{3,4} T 3 , 4
Here Δ ( T 3 , 4 ) = t − 3 − t − 2 + 1 − t 2 + t 3 \Delta(T_{3,4})=t^{-3}-t^{-2}+1-t^{2}+t^{3} Δ ( T 3 , 4 ) = t − 3 − t − 2 + 1 − t 2 + t 3 1 , 2 , 2 , 1 1,2,2,1 1 , 2 , 2 , 1 C F K U , V ( T 3 , 4 ) CFK_{U,V}(T_{3,4}) C F K U , V ( T 3 , 4 )
x 1 ← U x 2 x 1 ↓ V x 3 ← U 2 x 4 x 3 ↓ V x 3 x 5 \begin{aligned}
x_{1}&\xleftarrow{U}x_{2}\\
&\phantom{x_{1}}\qquad\downarrow V\\
x_{3}&\xleftarrow{U^{2}}x_{4}\\
&\phantom{x_{3}}\qquad\downarrow V\\
&\phantom{x_{3}}\qquad x_{5}
\end{aligned} x 1 x 3 U x 2 x 1 ↓ V U 2 x 4 x 3 ↓ V x 3 x 5 Take n = 2 n=2 n = 2 V V V 2 U 2U 2 U U 2 = 0 U^2=0 U 2 = 0 C F K 2 ( T 3 , 4 ) CFK_2(T_{3,4}) C F K 2 ( T 3 , 4 ) ∂ 2 x 2 = U x 1 \partial_2x_2=Ux_1 ∂ 2 x 2 = U x 1 ∂ 2 x 4 = 2 U x 5 \partial_2x_4=2Ux_5 ∂ 2 x 4 = 2 U x 5 C F K − ( T 3 , 4 ) ⊗ Q [ U ] Q [ U ] ( U 2 ) CFK^-(T_{3,4})\otimes_{\mathbb{Q}[U]}\frac{\mathbb{Q}[U]}{(U^2)} C F K − ( T 3 , 4 ) ⊗ Q [ U ] ( U 2 ) Q [ U ] ∂ x 2 = U x 1 \partial x_2=Ux_1 ∂ x 2 = U x 1 ∂ x 4 = 0 \partial x_4=0 ∂ x 4 = 0
H F K 2 ( T 3 , 4 ) ≅ Q [ U ] ( U 2 ) ⊕ Q 4 . HFK_2(T_{3,4})\cong\frac{\mathbb{Q}[U]}{(U^2)}\oplus\mathbb{Q}^4. H F K 2 ( T 3 , 4 ) ≅ ( U 2 ) Q [ U ] ⊕ Q 4 . of dimension 6. On the other hand H F K − ( T 3 , 4 ) ≅ Q [ U ] ⊕ Q [ U ] ( U ) ⊕ Q [ U ] ( U 2 ) HFK^-(T_{3,4})\cong\mathbb{Q}[U]\oplus\frac{\mathbb{Q}[U]}{(U)}\oplus\frac{\mathbb{Q}[U]}{(U^2)} H F K − ( T 3 , 4 ) ≅ Q [ U ] ⊕ ( U ) Q [ U ] ⊕ ( U 2 ) Q [ U ]
H F K − ( T 3 , 4 ) ⊗ Q [ U ] Q [ U ] ( U 2 ) ⊕ Tor 1 Q [ U ] ( H F K − ( T 3 , 4 ) ; Q [ U ] ( U 2 ) ) ≅ ( Q [ U ] ( U 2 ) ) 3 ⊕ Q 2 . HFK^-(T_{3,4})\otimes_{\mathbb{Q}[U]}\frac{\mathbb{Q}[U]}{(U^2)}\oplus\operatorname{Tor}^{\mathbb{Q}[U]}_1\left(HFK^-(T_{3,4});\frac{\mathbb{Q}[U]}{(U^2)}\right)\cong\left(\frac{\mathbb{Q}[U]}{(U^2)}\right)^3\oplus\mathbb{Q}^2. H F K − ( T 3 , 4 ) ⊗ Q [ U ] ( U 2 ) Q [ U ] ⊕ Tor 1 Q [ U ] ( H F K − ( T 3 , 4 ) ; ( U 2 ) Q [ U ] ) ≅ ( ( U 2 ) Q [ U ] ) 3 ⊕ Q 2 . of dimension 8. The value 6 agrees with Dowlin’s computation dim Q H F K 2 ( T 3 , 3 k + 1 ) = 2 + 4 k \dim_{\mathbb{Q}}HFK_2(T_{3,3k+1})=2+4k dim Q H F K 2 ( T 3 , 3 k + 1 ) = 2 + 4 k [1 Lemma 5.9] . The discrepancy comes from the arrow x 4 → 2 U x 5 x_4\to2Ux_5 x 4 → 2 U x 5 C F K 2 CFK_2 C F K 2 n − 1 = 1 n-1=1 n − 1 = 1 x 4 x_4 x 4 Lemma 4.13 and Proposition 4.15 ). For n ≥ 3 n\geq3 n ≥ 3 n + 6 n+6 n + 6
Proof of Main Theorem
Motivated by the preceding examples, we prove Theorem 1.4 using the universal coefficient theorem. In fact, since Q [ U ] \mathbb{Q}[U] Q [ U ] C F K − ( K ) CFK^-(K) C F K − ( K ) Q [ U ] \mathbb{Q}[U] Q [ U ] [19 ] (see also [4 Theorem 6.1] ),
H ∗ ( C F K − ( K ) ⊗ Q [ U ] Q [ U ] ( U n ) ) ≅ [ H F K − ( K ) ⊗ Q [ U ] Q [ U ] ( U n ) ] ⊕ Tor 1 Q [ U ] ( H F K − ( K ) ; Q [ U ] ( U n ) ) . H_*\left(CFK^-(K)\otimes_{\mathbb{Q}[U]}\frac{\mathbb{Q}[U]}{(U^n)}\right)\cong\left[HFK^-(K)\otimes_{\mathbb{Q}[U]}\frac{\mathbb{Q}[U]}{(U^n)}\right]\oplus\operatorname{Tor}^{\mathbb{Q}[U]}_1\left(HFK^-(K);\frac{\mathbb{Q}[U]}{(U^n)}\right). H ∗ ( C F K − ( K ) ⊗ Q [ U ] ( U n ) Q [ U ] ) ≅ [ H F K − ( K ) ⊗ Q [ U ] ( U n ) Q [ U ] ] ⊕ Tor 1 Q [ U ] ( H F K − ( K ) ; ( U n ) Q [ U ] ) . Here and below these are isomorphisms of ungraded Q [ U ] \mathbb{Q}[U] Q [ U ]
Recall that, by Theorem 2.12 , Dowlin’s complex C F K n ( K ) CFK_n(K) C F K n ( K )
C F K n ( K ) ≃ C F K U , V ( K ) ( U V , V − n U n − 1 ) . CFK_n(K)\simeq\frac{CFK_{U,V}(K)}{(UV,V-nU^{n-1})}. C F K n ( K ) ≃ ( U V , V − n U n − 1 ) C F K U , V ( K ) . In Examples 3.2 and 3.3 in § 3 , for n ≥ 2 n\geq2 n ≥ 2
C F K − ( K ) ⊗ Q [ U ] Q [ U ] ( U n ) CFK^-(K)\otimes_{\mathbb{Q}[U]}\frac{\mathbb{Q}[U]}{(U^n)} C F K − ( K ) ⊗ Q [ U ] ( U n ) Q [ U ] which we can compute using the universal coefficient theorem. This motivates us to propose another definition:
Definition 4.1. Given a finitely generated free chain complex C = ⨁ Q [ U , V ] C=\bigoplus\mathbb{Q}[U,V] C = ⨁ Q [ U , V ] Q [ U , V ] \mathbb{Q}[U,V] Q [ U , V ]
C n : = C ( U V , V − n U n − 1 ) C_n:=\frac{C}{(UV,V-nU^{n-1})} C n := ( U V , V − n U n − 1 ) C C ~ n : = C ( V , U n ) . \widetilde{C}_n:=\frac{C}{(V,U^n)}. C n := ( V , U n ) C . Remark 4.2. Notice that
C ( U V , V − n U n − 1 ) ≅ C ⊗ Q [ U , V ] Q [ U , V ] ( U V , V − n U n − 1 ) , \frac{C}{(UV,V-nU^{n-1})}\cong C\otimes_{\mathbb{Q}[U,V]}\frac{\mathbb{Q}[U,V]}{(UV,V-nU^{n-1})}, ( U V , V − n U n − 1 ) C ≅ C ⊗ Q [ U , V ] ( U V , V − n U n − 1 ) Q [ U , V ] , and similarly that
C ( V , U n ) ≅ C ⊗ Q [ U , V ] Q [ U , V ] ( V , U n ) ≅ C ( V ) ⊗ Q [ U ] Q [ U ] ( U n ) , \frac{C}{(V,U^n)}\cong C\otimes_{\mathbb{Q}[U,V]}\frac{\mathbb{Q}[U,V]}{(V,U^n)}\cong\frac{C}{(V)}\otimes_{\mathbb{Q}[U]}\frac{\mathbb{Q}[U]}{(U^n)}, ( V , U n ) C ≅ C ⊗ Q [ U , V ] ( V , U n ) Q [ U , V ] ≅ ( V ) C ⊗ Q [ U ] ( U n ) Q [ U ] , so C F K n ( K ) CFK_n(K) C F K n ( K ) C F K ~ n ( K ) : = C F K U , V ( K ) / ( V , U n ) ≅ C F K − ( K ) ⊗ Q [ U ] ( U n ) \widetilde{CFK}_n(K):=CFK_{U,V}(K)/(V,U^n)\cong CFK^-(K)\otimes\frac{\mathbb{Q}[U]}{(U^n)} C F K n ( K ) := C F K U , V ( K ) / ( V , U n ) ≅ C F K − ( K ) ⊗ ( U n ) Q [ U ]
In general, C n C_n C n C ~ n \widetilde{C}_n C n Q [ U , V ] \mathbb{Q}[U,V] Q [ U , V ] Q [ U ] \mathbb{Q}[U] Q [ U ] Remark 3.1 ). In fact, we have the following counterexample:
Lemma 4.3. C n C_n C n C ~ n \widetilde{C}_n C n
Proof. Consider the complex C C C
Q [ U , V ] → V Q [ U , V ] . \mathbb{Q}[U,V]\xrightarrow{V}\mathbb{Q}[U,V]. Q [ U , V ] V Q [ U , V ] . Then C n C_n C n
Q [ U ] ( U n ) → n U n − 1 Q [ U ] ( U n ) . \frac{\mathbb{Q}[U]}{(U^n)}\xrightarrow{nU^{n-1}}\frac{\mathbb{Q}[U]}{(U^n)}. ( U n ) Q [ U ] n U n − 1 ( U n ) Q [ U ] . and C ~ n \widetilde{C}_n C n
Q [ U ] ( U n ) → 0 Q [ U ] ( U n ) . \frac{\mathbb{Q}[U]}{(U^n)}\xrightarrow{0}\frac{\mathbb{Q}[U]}{(U^n)}. ( U n ) Q [ U ] 0 ( U n ) Q [ U ] . On the level of homology,
H ∗ ( C n ) ≅ Q [ U ] ( U n − 1 ) ⊕ Q [ U ] ( U n − 1 ) . H_*(C_n)\cong\frac{\mathbb{Q}[U]}{(U^{n-1})}\oplus\frac{\mathbb{Q}[U]}{(U^{n-1})}. H ∗ ( C n ) ≅ ( U n − 1 ) Q [ U ] ⊕ ( U n − 1 ) Q [ U ] . H ∗ ( C ~ n ) ≅ Q [ U ] ( U n ) ⊕ Q [ U ] ( U n ) . H_*(\widetilde{C}_n)\cong\frac{\mathbb{Q}[U]}{(U^n)}\oplus\frac{\mathbb{Q}[U]}{(U^n)}. H ∗ ( C n ) ≅ ( U n ) Q [ U ] ⊕ ( U n ) Q [ U ] . Although a general complex over Q [ U , V ] \mathbb{Q}[U,V] Q [ U , V ]
L-space and thin knots
The aforementioned classes of knots of interest are L L L § 1 ; throughout, thinness refers to H F K ^ ( K ; Q ) \widehat{HFK}(K;\mathbb{Q}) H F K ( K ; Q )
Definition 4.4 (Box). A chain complex C = ⨁ i = 1 4 Q [ U , V ] ⟨ x i ⟩ C=\bigoplus_{i=1}^{4}\mathbb{Q}[U,V]\langle x_i\rangle C = ⨁ i = 1 4 Q [ U , V ] ⟨ x i ⟩
x 2 ← U k x 1 V l ↓ ↓ V l x 4 ← U k x 3 \begin{CD}
x_2 @<{U^k}<< x_1 \\
@V{V^l}VV @VV{V^l}V \\
x_4 @<{U^k}<< x_3
\end{CD} x 2 V l ↓ ⏐ x 4 U k U k x 1 ↓ ⏐ V l x 3 for some positive integers k k k l l l x i x_i x i Q [ U , V ] \mathbb{Q}[U,V] Q [ U , V ] i i i C C C
Definition 4.5 (Staircase). A chain complex C = ⨁ i = 1 m Q [ U , V ] ⟨ x i ⟩ C=\bigoplus_{i=1}^{m}\mathbb{Q}[U,V]\langle x_i\rangle C = ⨁ i = 1 m Q [ U , V ] ⟨ x i ⟩ Figure 4 for some positive integers k i k_i k i m ≥ 1 m\ge1 m ≥ 1 x i x_i x i Q [ U , V ] \mathbb{Q}[U,V] Q [ U , V ] i i i C C C x i x_i x i x i + 1 x_{i+1} x i + 1 k i k_i k i x i x_i x i i i i x i x_i x i i i i x i x_i x i
∂ x i = U k i − 1 x i − 1 + V k i x i + 1 , \partial x_i=U^{k_i-1}x_{i-1}+V^{k_i}x_{i+1}, ∂ x i = U k i − 1 x i − 1 + V k i x i + 1 , where terms involving x 0 x_0 x 0 x m + 1 x_{m+1} x m + 1 m = 1 m=1 m = 1 C = Q [ U , V ] ⟨ x 1 ⟩ C=\mathbb{Q}[U,V]\langle x_1\rangle C = Q [ U , V ] ⟨ x 1 ⟩
Figure 4. The two possible staircase complexes: the left form and the right form.
Example 4.6. The master complex for the trefoil, C F K U , V ( T 2 , 3 ) CFK_{U,V}(T_{2,3}) C F K U , V ( T 2 , 3 ) Example 3.2 , is a staircase complex.
Figure 5. The tower (vector space) structure of the complex C F K n ( T 2 , 3 ) CFK_n(T_{2,3}) C F K n ( T 2 , 3 ) Figure 2 ). For example, ∂ n ( x 2 ) = U x 1 + n U n − 1 x 3 \partial_n(x_2)=Ux_1+nU^{n-1}x_3 ∂ n ( x 2 ) = U x 1 + n U n − 1 x 3 ∂ n ( U x 2 ) = U 2 x 1 \partial_n(Ux_2)=U^2x_1 ∂ n ( U x 2 ) = U 2 x 1
With these definitions in place, we appeal to two structure theorems concerning the master complex C F K U , V ( K ) CFK_{U,V}(K) C F K U , V ( K ) L L L Q [ U , V ] \mathbb{Q}[U,V] Q [ U , V ] Theorem 4.14 rely on the following Lemma:
Lemma 4.8 (Cancellation lemma). [4 Lemma 4.1] Let ( C , ∂ ) (C,\partial) ( C , ∂ ) R R R { x i } \{x_i\} { x i } d ( x k , x l ) d(x_k,x_l) d ( x k , x l ) R R R
∂ ( x k ) = ∑ m ≠ l a m x m + d ( x k , x l ) x l . \partial(x_k)=\sum_{m\ne l}a_mx_m+d(x_k,x_l)x_l. ∂ ( x k ) = m = l ∑ a m x m + d ( x k , x l ) x l . Then we can define a complex ( C ′ , ∂ ′ ) (C',\partial') ( C ′ , ∂ ′ ) { x i ∣ i ≠ k , i ≠ l } \{x_i\mid i\ne k,\ i\ne l\} { x i ∣ i = k , i = l } ( C , ∂ ) (C,\partial) ( C , ∂ ) d ( x i , x j ) d(x_i,x_j) d ( x i , x j ) x j x_j x j ∂ x i \partial x_i ∂ x i d ′ ( x i , x j ) = d ( x i , x j ) − d ( x i , x l ) d ( x k , x l ) − 1 d ( x k , x j ) d'(x_i,x_j)=d(x_i,x_j)-d(x_i,x_l)d(x_k,x_l)^{-1}d(x_k,x_j) d ′ ( x i , x j ) = d ( x i , x j ) − d ( x i , x l ) d ( x k , x l ) − 1 d ( x k , x j )
Theorem 4.9. (cf. [15 Lemma 7] ) If a knot K K K C F K U , V ( K ) CFK_{U,V}(K) C F K U , V ( K ) Q [ U , V ] \mathbb{Q}[U,V] Q [ U , V ] k i = 1 k_i=1 k i = 1 m = 2 ∣ τ ( K ) ∣ + 1 m=2|\tau(K)|+1 m = 2∣ τ ( K ) ∣ + 1 Figure 4 if τ ( K ) ≥ 0 \tau(K)\ge0 τ ( K ) ≥ 0 τ ( K ) < 0 \tau(K)<0 τ ( K ) < 0 k = l = 1 k=l=1 k = l = 1
Proof. Here we need the gradings that were suppressed in § 2 . The master complex C F K U , V ( K ) CFK_{U,V}(K) C F K U , V ( K ) ( gr U , gr V ) (\operatorname{gr}_U,\operatorname{gr}_V) ( gr U , gr V ) U U U ( − 2 , 0 ) (-2,0) ( − 2 , 0 ) V V V ( 0 , − 2 ) (0,-2) ( 0 , − 2 ) ( − 1 , − 1 ) (-1,-1) ( − 1 , − 1 ) A = 1 2 ( gr U − gr V ) A=\frac{1}{2}(\operatorname{gr}_U-\operatorname{gr}_V) A = 2 1 ( gr U − gr V ) M = gr U M=\operatorname{gr}_U M = gr U
Step 1: every arrow has length one. Applying Lemma 4.8 repeatedly to arrows whose coefficients are invertible, we obtain a complex C C C C F K U , V ( K ) CFK_{U,V}(K) C F K U , V ( K ) Q [ U , V ] \mathbb{Q}[U,V] Q [ U , V ] ∂ C ⊂ ( U , V ) C \partial C\subset(U,V)C ∂ C ⊂ ( U , V ) C Ω \Omega Ω Q \mathbb{Q} Q A A A Ω = ⨁ a Ω a \Omega=\bigoplus_a\Omega_a Ω = ⨁ a Ω a C / ( U , V ) C/(U,V) C / ( U , V ) Ω ≅ H F K ^ ( K ; Q ) \Omega\cong\widehat{HFK}(K;\mathbb{Q}) Ω ≅ H F K ( K ; Q ) K K K A − M = − 1 2 ( gr U + gr V ) A-M=-\frac{1}{2}(\operatorname{gr}_U+\operatorname{gr}_V) A − M = − 2 1 ( gr U + gr V ) Ω \Omega Ω c U i V j y cU^iV^jy c U i V j y c ≠ 0 c\ne0 c = 0 ∂ x \partial x ∂ x x x x y y y gr U + gr V \operatorname{gr}_U+\operatorname{gr}_V gr U + gr V
( gr U + gr V ) ( y ) − 2 ( i + j ) = ( gr U + gr V ) ( x ) − 2 , (\operatorname{gr}_U+\operatorname{gr}_V)(y)-2(i+j)=(\operatorname{gr}_U+\operatorname{gr}_V)(x)-2, ( gr U + gr V ) ( y ) − 2 ( i + j ) = ( gr U + gr V ) ( x ) − 2 , so i + j = 1 i+j=1 i + j = 1 Q \mathbb{Q} Q ξ , η : Ω → Ω \xi,\eta:\Omega\to\Omega ξ , η : Ω → Ω
∂ = U ξ + V η \partial=U\xi+V\eta ∂ = U ξ + V η on C = Ω ⊗ Q [ U , V ] C=\Omega\otimes\mathbb{Q}[U,V] C = Ω ⊗ Q [ U , V ] ξ ( Ω a ) ⊂ Ω a + 1 \xi(\Omega_a)\subset\Omega_{a+1} ξ ( Ω a ) ⊂ Ω a + 1 η ( Ω a ) ⊂ Ω a − 1 \eta(\Omega_a)\subset\Omega_{a-1} η ( Ω a ) ⊂ Ω a − 1
Step 2: an exterior algebra. Expanding ∂ 2 = U 2 ξ 2 + U V ( ξ η + η ξ ) + V 2 η 2 = 0 \partial^2=U^2\xi^2+UV(\xi\eta+\eta\xi)+V^2\eta^2=0 ∂ 2 = U 2 ξ 2 + U V ( ξ η + η ξ ) + V 2 η 2 = 0
ξ 2 = 0 , η 2 = 0 , ξ η = − η ξ . (1) \xi^2=0,\qquad\eta^2=0,\qquad\xi\eta=-\eta\xi. \tag*{(1)} ξ 2 = 0 , η 2 = 0 , ξ η = − η ξ . (1) So Ω \Omega Ω Λ = Q ⟨ ξ , η ⟩ / ( ξ 2 , η 2 , ξ η + η ξ ) \Lambda= \mathbb{Q}\langle\xi,\eta\rangle/(\xi^2,\eta^2,\xi\eta+\eta\xi) Λ = Q ⟨ ξ , η ⟩ / ( ξ 2 , η 2 , ξ η + η ξ ) Ω = Ω ′ ⊕ Ω ′ ′ \Omega=\Omega^\prime\oplus\Omega^{\prime\prime} Ω = Ω ′ ⊕ Ω ′′ Λ \Lambda Λ
C = ( Ω ′ ⊗ Q [ U , V ] ) ⊕ ( Ω ′ ′ ⊗ Q [ U , V ] ) . C=(\Omega^\prime\otimes\mathbb{Q}[U,V])\oplus(\Omega^{\prime\prime}\otimes\mathbb{Q}[U,V]). C = ( Ω ′ ⊗ Q [ U , V ]) ⊕ ( Ω ′′ ⊗ Q [ U , V ]) . Moreover, the complex ( Ω , ξ ) (\Omega,\xi) ( Ω , ξ ) C ⊗ Q [ U , V ] ( U − 1 , V − 1 ) ≃ C F K − ( K ) ⊗ Q [ U ] Q [ U ] ( U − 1 ) C\otimes\frac{\mathbb{Q}[U,V]}{(U^{-1},V^{-1})}\simeq CFK^-(K)\otimes_{\mathbb{Q}[U]}\frac{\mathbb{Q}[U]}{(U^{-1})} C ⊗ ( U − 1 , V − 1 ) Q [ U , V ] ≃ C F K − ( K ) ⊗ Q [ U ] ( U − 1 ) Q [ U ] H F ^ ( S 3 ) ≅ Q \widehat{HF}(S^3)\cong\mathbb{Q} H F ( S 3 ) ≅ Q [12 ] . In the same way ( Ω , η ) (\Omega,\eta) ( Ω , η ) H F ^ ( S 3 ) \widehat{HF}(S^3) H F ( S 3 ) w w w
dim Q H ∗ ( Ω , ξ ) = dim Q H ∗ ( Ω , η ) = 1. \dim_{\mathbb{Q}}H_*(\Omega,\xi)=\dim_{\mathbb{Q}}H_*(\Omega,\eta)=1. dim Q H ∗ ( Ω , ξ ) = dim Q H ∗ ( Ω , η ) = 1. Step 3: splitting off boxes. Suppose ξ η x ≠ 0 \xi\eta x\ne0 ξ η x = 0 x ∈ Ω a x\in\Omega_a x ∈ Ω a x x x ξ x \xi x ξ x η x \eta x η x ξ η x \xi\eta x ξ η x ξ x \xi x ξ x η x \eta x η x Ω a + 1 \Omega_{a+1} Ω a + 1 Ω a − 1 \Omega_{a-1} Ω a − 1 ξ η x = c x \xi\eta x=cx ξ η x = c x c 2 x = ( ξ η ) 2 x = − ξ 2 η 2 x = 0 c^2x=(\xi\eta)^2x=-\xi^2\eta^2x=0 c 2 x = ( ξ η ) 2 x = − ξ 2 η 2 x = 0 Γ \Gamma Γ Λ \Lambda Λ λ : Ω → Q \lambda:\Omega\to\mathbb{Q} λ : Ω → Q Ω a \Omega_a Ω a λ ( x ) = 0 \lambda(x)=0 λ ( x ) = 0 λ ( ξ η x ) = 1 \lambda(\xi\eta x)=1 λ ( ξ η x ) = 1 ρ : Ω → Γ \rho:\Omega\to\Gamma ρ : Ω → Γ
ρ ( w ) = λ ( ξ η w ) x − λ ( η w ) ξ x + λ ( ξ w ) η x + λ ( w ) ξ η x . \rho(w)=\lambda(\xi\eta w)x-\lambda(\eta w)\xi x+\lambda(\xi w)\eta x+\lambda(w)\xi\eta x. ρ ( w ) = λ ( ξ η w ) x − λ ( η w ) ξ x + λ ( ξ w ) η x + λ ( w ) ξ η x . A direct check using the relations of Step 2 shows that ρ ξ = ξ ρ \rho\xi=\xi\rho ρ ξ = ξ ρ ρ η = η ρ \rho\eta=\eta\rho ρ η = η ρ λ \lambda λ Ω a \Omega_a Ω a ρ \rho ρ ρ ( x ) = x \rho(x)=x ρ ( x ) = x ρ \rho ρ Γ \Gamma Γ Ω = Γ ⊕ ker ρ \Omega=\Gamma\oplus\ker\rho Ω = Γ ⊕ ker ρ Λ \Lambda Λ
x 1 = x , x 2 = ξ x , x 3 = − η x , x 4 = η ξ x , x_1=x,\qquad x_2=\xi x,\qquad x_3=-\eta x,\qquad x_4=\eta\xi x, x 1 = x , x 2 = ξ x , x 3 = − η x , x 4 = η ξ x , the complex Γ ⊗ Q [ U , V ] \Gamma\otimes\mathbb{Q}[U,V] Γ ⊗ Q [ U , V ] Definition 4.4 with k = l = 1 k=l=1 k = l = 1 ∂ x 1 = U x 2 − V x 3 \partial x_1=Ux_2-Vx_3 ∂ x 1 = U x 2 − V x 3 ∂ x 2 = V x 4 \partial x_2=Vx_4 ∂ x 2 = V x 4 ∂ x 3 = U x 4 \partial x_3=Ux_4 ∂ x 3 = U x 4 dim Q Ω \dim_{\mathbb{Q}}\Omega dim Q Ω
Ω = Γ 1 ⊕ ⋯ ⊕ Γ b ⊕ Ω ′ , \Omega=\Gamma_1\oplus\cdots\oplus\Gamma_b\oplus\Omega^\prime, Ω = Γ 1 ⊕ ⋯ ⊕ Γ b ⊕ Ω ′ , where each Γ i ⊗ Q [ U , V ] \Gamma_i\otimes\mathbb{Q}[U,V] Γ i ⊗ Q [ U , V ] k = l = 1 k=l=1 k = l = 1 ξ η = η ξ = 0 \xi\eta=\eta\xi=0 ξ η = η ξ = 0 Ω ′ \Omega^\prime Ω ′
Step 4: staircases. Let Θ = ker ξ ∩ ker η ⊂ Ω ′ \Theta=\ker\xi\cap\ker\eta\subset\Omega^\prime Θ = ker ξ ∩ ker η ⊂ Ω ′ Ψ \Psi Ψ Θ \Theta Θ Ω ′ \Omega^\prime Ω ′ ξ \xi ξ η \eta η Ω ′ \Omega^\prime Ω ′ Ω ′ \Omega^\prime Ω ′ Θ \Theta Θ Θ \Theta Θ Ω ′ \Omega^\prime Ω ′ ξ : Ψ a → Θ a + 1 \xi:\Psi_a\to\Theta_{a+1} ξ : Ψ a → Θ a + 1 η : Ψ a → Θ a − 1 \eta:\Psi_a\to\Theta_{a-1} η : Ψ a → Θ a − 1 Ψ a \Psi_a Ψ a Θ a \Theta_a Θ a a ∈ Z a\in\mathbb{Z} a ∈ Z Ω ′ \Omega^\prime Ω ′ Λ \Lambda Λ
⋯ − Θ a − 1 − Ψ a − Θ a + 1 − Ψ a + 2 − ⋯ , \cdots-\Theta_{a-1}-\Psi_a-\Theta_{a+1}-\Psi_{a+2}-\cdots, ⋯ − Θ a − 1 − Ψ a − Θ a + 1 − Ψ a + 2 − ⋯ , and since Ω ′ \Omega^\prime Ω ′ [3 ] , it is a direct sum of interval representations, which are one-dimensional at each vertex of a segment of a line, zero elsewhere, and have every arrow in the segment an isomorphism. After rescaling the basis vectors along the segment, each such summand Υ \Upsilon Υ Υ ⊗ Q [ U , V ] \Upsilon\otimes\mathbb{Q}[U,V] Υ ⊗ Q [ U , V ] k i = 1 k_i=1 k i = 1 Ψ \Psi Ψ Θ \Theta Θ m = 1 m=1 m = 1 Ψ \Psi Ψ
Step 5: exactly one staircase. The complexes ( Γ i , ξ ) (\Gamma_i,\xi) ( Γ i , ξ ) ( Γ i , η ) (\Gamma_i,\eta) ( Γ i , η ) Υ \Upsilon Υ v v v v − 1 v-1 v − 1 e ξ e_\xi e ξ ξ \xi ξ e η e_\eta e η η \eta η
dim Q H ∗ ( Υ , ξ ) + dim Q H ∗ ( Υ , η ) = ( v − 2 e ξ ) + ( v − 2 e η ) = 2. \dim_{\mathbb{Q}}H_*(\Upsilon,\xi)+\dim_{\mathbb{Q}}H_*(\Upsilon,\eta)=(v-2e_\xi)+(v-2e_\eta)=2. dim Q H ∗ ( Υ , ξ ) + dim Q H ∗ ( Υ , η ) = ( v − 2 e ξ ) + ( v − 2 e η ) = 2. By Step 2 the total of these quantities over Ω \Omega Ω 2 2 2 Υ \Upsilon Υ dim Q H ∗ ( Υ , ξ ) = dim Q H ∗ ( Υ , η ) = 1 \dim_{\mathbb{Q}}H_*(\Upsilon,\xi)=\dim_{\mathbb{Q}}H_*(\Upsilon,\eta)=1 dim Q H ∗ ( Υ , ξ ) = dim Q H ∗ ( Υ , η ) = 1 e ξ = e η e_\xi=e_\eta e ξ = e η m = v = 2 e ξ + 1 m=v=2e_\xi+1 m = v = 2 e ξ + 1
C = S ⊕ B 1 ⊕ ⋯ ⊕ B b , \mathcal{C}=\mathcal{S}\oplus\mathcal{B}_1\oplus\cdots\oplus\mathcal{B}_b, C = S ⊕ B 1 ⊕ ⋯ ⊕ B b , where S = Υ ⊗ Q [ U , V ] \mathcal{S}=\Upsilon\otimes\mathbb{Q}[U,V] S = Υ ⊗ Q [ U , V ] k i = 1 k_i=1 k i = 1 B i = Γ i ⊗ Q [ U , V ] \mathcal{B}_i=\Gamma_i\otimes\mathbb{Q}[U,V] B i = Γ i ⊗ Q [ U , V ] k = l = 1 k=l=1 k = l = 1
Step 6: the length of the staircase. We use the convention [6 ]
τ ( K ) = − max { A ( y ) : y ∈ H F K − ( K ) homogeneous , U d y ≠ 0 for all d } . \tau(K)=-\max\{A(y):y\in HFK^-(K)\text{ homogeneous},\ U^d y\ne0\text{ for all }d\}. τ ( K ) = − max { A ( y ) : y ∈ H F K − ( K ) homogeneous , U d y = 0 for all d } . Number the generators of S \mathcal{S} S Definition 4.5 . Since ξ \xi ξ η \eta η A A A A ( x j ) = A ( x 1 ) − ( j − 1 ) A(x_j)=A(x_1)-(j-1) A ( x j ) = A ( x 1 ) − ( j − 1 ) B i / ( V ) \mathcal{B}_i/(V) B i / ( V ) Q ⟨ x 2 ⟩ ⊕ Q ⟨ x 4 ⟩ \mathbb{Q}\langle x_2\rangle\oplus\mathbb{Q}\langle x_4\rangle Q ⟨ x 2 ⟩ ⊕ Q ⟨ x 4 ⟩ U U U U U U H F K − ( K ) = H ∗ ( C / ( V ) ) HFK^-(K)=H_*(\mathcal{C}/(V)) H F K − ( K ) = H ∗ ( C / ( V )) S / ( V ) \mathcal{S}/(V) S / ( V ) x m x_m x m x 1 x_1 x 1 − τ ( K ) -\tau(K) − τ ( K ) w w w z z z [12 Section 3.5] interchanges U U U V V V A A A − A -A − A V V V H ∗ ( C / ( U ) ) H_*(\mathcal{C}/(U)) H ∗ ( C / ( U )) τ ( K ) \tau(K) τ ( K ) x 1 x_1 x 1 x m x_m x m 2 τ ( K ) = A ( x 1 ) − A ( x m ) = m − 1 2\tau(K)=A(x_1)-A(x_m)=m-1 2 τ ( K ) = A ( x 1 ) − A ( x m ) = m − 1 − 2 τ ( K ) = A ( x 1 ) − A ( x m ) = m − 1 -2\tau(K)=A(x_1)-A(x_m)=m-1 − 2 τ ( K ) = A ( x 1 ) − A ( x m ) = m − 1 m = 2 ∣ τ ( K ) ∣ + 1 m=2|\tau(K)|+1 m = 2∣ τ ( K ) ∣ + 1 τ ( K ) ≥ 0 \tau(K)\geq0 τ ( K ) ≥ 0 τ ( K ) < 0 \tau(K)<0 τ ( K ) < 0
Example 4.10 (The knot 5 2 5_2 5 2 5 2 5_2 5 2
Δ 5 2 ( t ) = 2 t − 3 + 2 t − 1 \Delta_{5_2}(t)=2t-3+2t^{-1} Δ 5 2 ( t ) = 2 t − 3 + 2 t − 1 and det ( 5 2 ) = 7 \det(5_2)=7 det ( 5 2 ) = 7 τ ( 5 2 ) = 1 \tau(5_2)=1 τ ( 5 2 ) = 1 H F K ^ ( 5 2 ; Q ) \widehat{HFK}(5_2;\mathbb{Q}) H F K ( 5 2 ; Q ) − 1 -1 − 1 M = A − 1 M=A-1 M = A − 1 C C C y 1 , y 2 y_1,y_2 y 1 , y 2 y 3 , y 4 , y 5 y_3,y_4,y_5 y 3 , y 4 , y 5 y 6 , y 7 y_6,y_7 y 6 , y 7 − 1 -1 − 1
∂ y 3 = U y 1 + V y 7 , ∂ y 4 = U y 2 − V y 6 + 2 V y 7 , ∂ y 2 = V y 5 , ∂ y 6 = U y 5 . \partial y_3=Uy_1+Vy_7,\qquad\partial y_4=Uy_2-Vy_6+2Vy_7,\qquad\partial y_2=Vy_5,\qquad\partial y_6=Uy_5. ∂ y 3 = U y 1 + V y 7 , ∂ y 4 = U y 2 − V y 6 + 2 V y 7 , ∂ y 2 = V y 5 , ∂ y 6 = U y 5 . with ∂ y 1 = ∂ y 5 = ∂ y 7 = 0 \partial y_1=\partial y_5=\partial y_7=0 ∂ y 1 = ∂ y 5 = ∂ y 7 = 0 C F K U , V ( 5 2 ) CFK_{U,V}(5_2) C F K U , V ( 5 2 )
ξ ( y 3 ) = y 1 , ξ ( y 4 ) = y 2 , ξ ( y 6 ) = y 5 , η ( y 2 ) = y 5 , η ( y 3 ) = y 7 , η ( y 4 ) = − y 6 + 2 y 7 , \xi(y_3)=y_1,\qquad\xi(y_4)=y_2,\qquad\xi(y_6)=y_5,\qquad\eta(y_2)=y_5,\qquad\eta(y_3)=y_7,\qquad\eta(y_4)=-y_6+2y_7, ξ ( y 3 ) = y 1 , ξ ( y 4 ) = y 2 , ξ ( y 6 ) = y 5 , η ( y 2 ) = y 5 , η ( y 3 ) = y 7 , η ( y 4 ) = − y 6 + 2 y 7 , and ξ \xi ξ η \eta η ξ \xi ξ η \eta η dim Q H ∗ ( Ω , ξ ) = dim Q H ∗ ( Ω , η ) = 7 − 6 = 1 \dim_{\mathbb{Q}}H_*(\Omega,\xi)=\dim_{\mathbb{Q}}H_*(\Omega,\eta)=7-6=1 dim Q H ∗ ( Ω , ξ ) = dim Q H ∗ ( Ω , η ) = 7 − 6 = 1
For Step 3, ξ η ( y 4 ) = ξ ( − y 6 + 2 y 7 ) = − y 5 ≠ 0 \xi\eta(y_4)=\xi(-y_6+2y_7)=-y_5\ne0 ξ η ( y 4 ) = ξ ( − y 6 + 2 y 7 ) = − y 5 = 0 x = y 4 x=y_4 x = y 4 λ \lambda λ λ ( y 5 ) = − 1 \lambda(y_5)=-1 λ ( y 5 ) = − 1 ρ \rho ρ y 2 , y 4 y_2,y_4 y 2 , y 4 y 5 y_5 y 5 y 6 y_6 y 6 y 6 − 2 y 7 y_6-2y_7 y 6 − 2 y 7 y 1 , y 3 y_1,y_3 y 1 , y 3 y 7 y_7 y 7
x 1 = y 4 , x 2 = y 2 , x 3 = y 6 − 2 y 7 , x 4 = y 5 . x_1=y_4,\qquad x_2=y_2,\qquad x_3=y_6-2y_7,\qquad x_4=y_5. x 1 = y 4 , x 2 = y 2 , x 3 = y 6 − 2 y 7 , x 4 = y 5 . and indeed ∂ x 1 = U x 2 − V x 3 \partial x_1=Ux_2-Vx_3 ∂ x 1 = U x 2 − V x 3 ∂ x 2 = V x 4 \partial x_2=Vx_4 ∂ x 2 = V x 4 ∂ x 3 = U x 4 \partial x_3=Ux_4 ∂ x 3 = U x 4 y 6 ↦ y 6 − 2 y 7 y_6\mapsto y_6-2y_7 y 6 ↦ y 6 − 2 y 7
For Step 4, Ω ′ = ker ρ \Omega'=\ker\rho Ω ′ = ker ρ y 1 , y 3 y_1,y_3 y 1 , y 3 y 7 y_7 y 7 Θ \Theta Θ y 1 y_1 y 1 y 7 y_7 y 7 Ψ \Psi Ψ y 3 y_3 y 3 Θ 1 − Ψ 0 − Θ − 1 \Theta_1-\Psi_0-\Theta_{-1} Θ 1 − Ψ 0 − Θ − 1 Θ \Theta Θ x 1 = y 1 x_1=y_1 x 1 = y 1 x 2 = y 3 x_2=y_3 x 2 = y 3 x 3 = y 7 x_3=y_7 x 3 = y 7 ∂ x 2 = U x 1 + V x 3 \partial x_2=Ux_1+Vx_3 ∂ x 2 = U x 1 + V x 3
C F K U , V ( 5 2 ) ≃ C F K U , V ( T 2 , 3 ) ⊕ B , CFK_{U,V}(5_2)\simeq CFK_{U,V}(T_{2,3})\oplus\mathcal{B}, C F K U , V ( 5 2 ) ≃ C F K U , V ( T 2 , 3 ) ⊕ B , where B \mathcal{B} B k = l = 1 k=l=1 k = l = 1 U U U H F K − ( 5 2 ) HFK^-(5_2) H F K − ( 5 2 ) x 3 = y 7 x_3=y_7 x 3 = y 7 − 1 = − τ ( 5 2 ) -1=-\tau(5_2) − 1 = − τ ( 5 2 ) m = 3 = 2 τ ( 5 2 ) + 1 m=3=2\tau(5_2)+1 m = 3 = 2 τ ( 5 2 ) + 1 H F K n ( 5 2 ) ≅ Q [ U ] ( U n ) ⊕ Q 6 HFK_n(5_2)\cong\frac{\mathbb{Q}[U]}{(U^n)}\oplus\mathbb{Q}^6 H F K n ( 5 2 ) ≅ ( U n ) Q [ U ] ⊕ Q 6 n ≥ 2 n\geq2 n ≥ 2
Theorem 4.11. [13 Theorem 1.2] , [5 Remark 6.6] If a knot K K K L L L C F K U , V ( K ) CFK_{U,V}(K) C F K U , V ( K ) Q [ U , V ] \mathbb{Q}[U,V] Q [ U , V ] Figure 4 , whose step lengths k 1 , … , k m − 1 k_1,\ldots,k_{m-1} k 1 , … , k m − 1 Δ K \Delta_K Δ K
Here [15 Lemma 7] proves the decomposition of Theorem 4.9 over F 2 \mathbb{F}_2 F 2 C F K − ( K ) CFK^-(K) C F K − ( K ) [13 Theorem 1.2] determines H F K ^ ( K ) \widehat{HFK}(K) H F K ( K ) L L L [5 Remark 6.6] ; over Q \mathbb{Q} Q
Main theorem
To prove the main theorem, we employ the following algebraic lemma along with Lemma 4.8. The second term in d ′ d' d ′
Lemma 4.12. Let n ≥ 2 n\geq2 n ≥ 2 C C C l ≥ 2 l\geq2 l ≥ 2 k ≤ n − 1 k\leq n-1 k ≤ n − 1 C n ≅ C ~ n C_n\cong\widetilde{C}_n C n ≅ C n Q [ U ] ( U n ) \frac{\mathbb{Q}[U]}{(U^n)} ( U n ) Q [ U ]
H ∗ ( C n ) ≅ H ∗ ( C ~ n ) ≅ ( Q [ U ] ( U min ( k , n ) ) ) 4 H_*(C_n)\cong H_*(\widetilde{C}_n)\cong\left(\frac{\mathbb{Q}[U]}{(U^{\min(k,n)})}\right)^4 H ∗ ( C n ) ≅ H ∗ ( C n ) ≅ ( ( U m i n ( k , n ) ) Q [ U ] ) 4 is an isomorphism of Q [ U ] \mathbb{Q}[U] Q [ U ] l = 1 l=1 l = 1 k ≥ n k\geq n k ≥ n
H ∗ ( C n ) ≅ ( Q [ U ] ( U n − 1 ) ) 4 , H ∗ ( C ~ n ) ≅ ( Q [ U ] ( U n ) ) 4 . H_*(C_n)\cong\left(\frac{\mathbb{Q}[U]}{(U^{n-1})}\right)^4,\qquad H_*(\widetilde{C}_n)\cong\left(\frac{\mathbb{Q}[U]}{(U^n)}\right)^4. H ∗ ( C n ) ≅ ( ( U n − 1 ) Q [ U ] ) 4 , H ∗ ( C n ) ≅ ( ( U n ) Q [ U ] ) 4 . Proof. Any box complex can be pictorially represented as in Definition 4.4 . Note that after the identifications V = n U n − 1 V=nU^{n-1} V = n U n − 1 U V = 0 UV=0 U V = 0 l > 1 l>1 l > 1 n ≥ 2 n\ge2 n ≥ 2
V l = V ⋅ V l − 1 = n U n − 1 V l − 1 = U V ⋅ n U n − 2 V l − 2 = 0. \begin{aligned}
V^l &= V\cdot V^{l-1}\\
&=nU^{n-1}V^{l-1}\\
&=UV\cdot nU^{n-2}V^{l-2}\\
&=0.
\end{aligned} V l = V ⋅ V l − 1 = n U n − 1 V l − 1 = U V ⋅ n U n − 2 V l − 2 = 0. So we see that if l > 1 l>1 l > 1 C n C_n C n C ~ n \widetilde{C}_n C n C n = C ~ n C_n=\widetilde{C}_n C n = C n l = 1 l=1 l = 1 C n C_n C n C ~ n \widetilde{C}_n C n Figure 6 . If k ≤ n − 1 k\le n-1 k ≤ n − 1 x 2 ′ = x 2 − n U n − 1 − k x 3 x'_2=x_2-nU^{n-1-k}x_3 x 2 ′ = x 2 − n U n − 1 − k x 3
Figure 6. On the left: C n C_n C n l = 1 l=1 l = 1 C ~ n \widetilde{C}_n C n l = 1 l=1 l = 1
∂ n x 1 = U k x 2 − n U n − 1 x 3 = U k x 2 ′ , ∂ n x 2 ′ = n U n − 1 x 4 − n U n − 1 − k U k x 4 = 0 , \partial_n x_1=U^k x_2-nU^{n-1}x_3=U^k x'_2,\qquad\partial_n x'_2=nU^{n-1}x_4-nU^{n-1-k}U^k x_4=0, ∂ n x 1 = U k x 2 − n U n − 1 x 3 = U k x 2 ′ , ∂ n x 2 ′ = n U n − 1 x 4 − n U n − 1 − k U k x 4 = 0 , so in the basis { x 1 , x 2 ′ , x 3 , x 4 } \{x_1,x'_2,x_3,x_4\} { x 1 , x 2 ′ , x 3 , x 4 } C n C_n C n C ~ n \widetilde{C}_n C n Figure 5 ,
H ∗ ( C n ) ≅ ( Q [ U ] ( U k ) ) 4 ≅ H ∗ ( C ~ n ) . H_*(C_n)\cong\left(\frac{\mathbb{Q}[U]}{(U^k)}\right)^4\cong H_*(\widetilde{C}_n). H ∗ ( C n ) ≅ ( ( U k ) Q [ U ] ) 4 ≅ H ∗ ( C n ) . If k ≥ n k\ge n k ≥ n U k = 0 U^k=0 U k = 0 C n C_n C n C ~ n \widetilde{C}_n C n
Lemma 4.13. Let n ≥ 2 n\ge2 n ≥ 2 C C C n − 1 n-1 n − 1
(1) if C C C k i − 1 ≤ n − 1 k_{i-1}\le n-1 k i − 1 ≤ n − 1 i i i
(2) if C C C k i + 1 ≤ n − 1 k_{i+1}\le n-1 k i + 1 ≤ n − 1 i ≤ m − 2 i\le m-2 i ≤ m − 2
Then C n ≅ C ~ n C_n\cong\widetilde{C}_n C n ≅ C n Q [ U ] ( U n ) \frac{\mathbb{Q}[U]}{(U^n)} ( U n ) Q [ U ] H ∗ ( C n ) ≅ H ∗ ( C ~ n ) H_*(C_n)\cong H_*(\widetilde{C}_n) H ∗ ( C n ) ≅ H ∗ ( C n ) Q [ U ] \mathbb{Q}[U] Q [ U ]
Proof. Any staircase complex can be pictorially represented as in Definition 4.5 . Using a similar argument as in the proof of Lemma 4.12 , every V k i V^{k_i} V k i k i ≥ 2 k_i\ge2 k i ≥ 2 C n C_n C n C n C_n C n C ~ n \widetilde{C}_n C n x i → n U n − 1 x i + 1 x_i\to nU^{n-1}x_{i+1} x i → n U n − 1 x i + 1 x i x_i x i k i = 1 k_i=1 k i = 1 ε i = 1 \varepsilon_i=1 ε i = 1 k i = 1 k_i=1 k i = 1 ε i = 0 \varepsilon_i=0 ε i = 0 x i x_i x i
∂ n x i = U k i − 1 x i − 1 + ε i n U n − 1 x i + 1 . \partial_n x_i=U^{k_i-1}x_{i-1}+\varepsilon_i nU^{n-1}x_{i+1}. ∂ n x i = U k i − 1 x i − 1 + ε i n U n − 1 x i + 1 . Rather than cancelling over Q \mathbb{Q} Q U U U Q [ U ] ( U n ) \frac{\mathbb{Q}[U]}{(U^n)} ( U n ) Q [ U ]
(1) For even i i i y i − 1 = x i − 1 + ε i n U n − 1 − k i − 1 x i + 1 y_{i-1}=x_{i-1}+\varepsilon_i nU^{n-1-k_{i-1}}x_{i+1} y i − 1 = x i − 1 + ε i n U n − 1 − k i − 1 x i + 1 y m = x m y_m=x_m y m = x m ∂ n x i = U k i − 1 y i − 1 \partial_n x_i=U^{k_{i-1}}y_{i-1} ∂ n x i = U k i − 1 y i − 1 Figure 7 . Since the odd generators are cycles, the Q [ U ] ( U n ) \frac{\mathbb{Q}[U]}{(U^n)} ( U n ) Q [ U ] C ~ n → C n \widetilde{C}_n\to C_n C n → C n x i x_i x i i i i x i ↦ y i x_i\mapsto y_i x i ↦ y i i i i
Figure 7. A local picture of the change of basis in the proof of Lemma 4.13 (left form) for some even i i i k i = 1 k_i = 1 k i = 1 k i − 1 ≤ n − 1 k_{i-1} \le n - 1 k i − 1 ≤ n − 1 C n C_n C n x i → n U n − 1 x i + 1 x_i \to nU^{n-1}x_{i+1} x i → n U n − 1 x i + 1 x i − 1 x_{i-1} x i − 1 y i − 1 = x i − 1 + n U n − 1 − k i − 1 x i + 1 y_{i-1} = x_{i-1} + nU^{n-1-k_{i-1}}x_{i+1} y i − 1 = x i − 1 + n U n − 1 − k i − 1 x i + 1 C ~ n \widetilde{C}_n C n
(2) Define recursively x m ′ = x m x'_m=x_m x m ′ = x m x i ′ = x i − ε i n U n − 1 − k i + 1 x i + 2 ′ x'_i=x_i-\varepsilon_i nU^{n-1-k_{i+1}}x'_{i+2} x i ′ = x i − ε i n U n − 1 − k i + 1 x i + 2 ′ i ≤ m − 2 i\le m-2 i ≤ m − 2 ∂ n x i ′ = U k i − 1 x i − 1 \partial_n x'_i=U^{k_{i-1}}x_{i-1} ∂ n x i ′ = U k i − 1 x i − 1 ∂ n x i + 2 ′ = U k i + 1 x i + 1 \partial_n x'_{i+2}=U^{k_{i+1}}x_{i+1} ∂ n x i + 2 ′ = U k i + 1 x i + 1 ε i n U n − 1 x i + 1 \varepsilon_i nU^{n-1}x_{i+1} ε i n U n − 1 x i + 1 C ~ n → C n \widetilde{C}_n\to C_n C n → C n x i ↦ x i ′ x_i\mapsto x'_i x i ↦ x i ′ i i i x i x_i x i i i i
We can readily compute H ∗ ( C n ) ≅ H ∗ ( C ~ n ) H_*(C_n)\cong H_*(\widetilde{C}_n) H ∗ ( C n ) ≅ H ∗ ( C n ) C ~ n \widetilde{C}_n C n m − 1 2 \frac{m-1}{2} 2 m − 1 Remark 4.7 ) with arrows given by the horizontal arrows, and one single tower with no arrows. For the left form, writing k s = min ( k 2 s − 1 , n ) k_s=\min(k_{2s-1},n) k s = min ( k 2 s − 1 , n )
H ∗ ( C n ) ≅ H ∗ ( C ~ n ) ≅ ⨁ s = 1 m − 1 2 ( Q [ U ] ( U k s ) ⟨ x 2 s − 1 ⟩ ⊕ Q [ U ] ( U k s ) ⟨ U n − k s x 2 s ⟩ ) ⊕ Q [ U ] ( U n ) ⟨ x m ⟩ , H_*(C_n)\cong H_*(\widetilde{C}_n)\cong\bigoplus_{s=1}^{\frac{m-1}{2}}\left(\frac{\mathbb{Q}[U]}{(U^{k_s})}_{\langle x_{2s-1}\rangle}\oplus\frac{\mathbb{Q}[U]}{(U^{k_s})}_{\langle U^{n-k_s}x_{2s}\rangle}\right)\oplus\frac{\mathbb{Q}[U]}{(U^n)}_{\langle x_m\rangle}, H ∗ ( C n ) ≅ H ∗ ( C n ) ≅ s = 1 ⨁ 2 m − 1 ( ( U k s ) Q [ U ] ⟨ x 2 s − 1 ⟩ ⊕ ( U k s ) Q [ U ] ⟨ U n − k s x 2 s ⟩ ) ⊕ ( U n ) Q [ U ] ⟨ x m ⟩ , and similarly for the right form, with κ s = min ( k 2 s , n ) \kappa_s = \min(k_{2s},n) κ s = min ( k 2 s , n ) x 2 s x_{2s} x 2 s U n − κ s x 2 s + 1 U^{n-\kappa_s}x_{2s+1} U n − κ s x 2 s + 1 x 1 x_1 x 1 Q [ U ] ( U n ) \frac{\mathbb{Q}[U]}{(U^n)} ( U n ) Q [ U ] Q [ U ] \mathbb{Q}[U] Q [ U ]
To conclude the main Theorem 1.4 , we make full use of the fact that, among a myriad of nice properties, all L L L
Theorem 4.14. Let n ≥ 2 n \ge2 n ≥ 2 K ⊂ S 3 K \subset S^3 K ⊂ S 3 L L L Δ K \Delta_K Δ K n − 1 n - 1 n − 1 Q [ U ] \mathbb{Q}[U] Q [ U ]
H F K n ( K ) ≅ ( H F K − ( K ) ⊗ Q [ U ] Q [ U ] ( U n ) ) ⊕ Tor 1 Q [ U ] ( H F K − ( K ) ; Q [ U ] ( U n ) ) . HF K_n(K) \cong\left(HFK^-(K) \mathbin{\otimes}_{\mathbb{Q}[U]} \frac{\mathbb{Q}[U]}{(U^n)}\right) \oplus\operatorname{Tor}^{\mathbb{Q}[U]}_1\left(HFK^-(K);\frac{\mathbb{Q}[U]}{(U^n)}\right). H F K n ( K ) ≅ ( H F K − ( K ) ⊗ Q [ U ] ( U n ) Q [ U ] ) ⊕ Tor 1 Q [ U ] ( H F K − ( K ) ; ( U n ) Q [ U ] ) . Proof. The argument is a chain of isomorphisms given by Lemmata 4.13, 4.12, and the universal coefficient theorem. Let C = C F K U , V ( K ) C = CFK_{U,V}(K) C = C F K U , V ( K ) Theorem 2.12 , H F K n ( K ) ≅ H ∗ ( C n ) HFK_n(K) \cong H_*(C_n) H F K n ( K ) ≅ H ∗ ( C n ) Q [ U ] \mathbb{Q}[U] Q [ U ] U V UV U V V = n U n − 1 V = nU^{n-1} V = n U n − 1 n U n nU^n n U n Q [ U ] \mathbb{Q}[U] Q [ U ] Theorems 4.9 and 4.11 , C C C Q [ U , V ] \mathbb{Q}[U,V] Q [ U , V ] C ↦ C n C \mapsto C_n C ↦ C n C ↦ C ~ n C \mapsto\widetilde{C}_n C ↦ C n Q [ U , V ] \mathbb{Q}[U,V] Q [ U , V ] k i = 1 ≤ n − 1 k_i = 1 \le n - 1 k i = 1 ≤ n − 1 k = l = 1 ≤ n − 1 k = l = 1 \le n - 1 k = l = 1 ≤ n − 1 L L L k i k_i k i Δ K \Delta_K Δ K n − 1 n - 1 n − 1
H ∗ ( C n ) ≅ H ∗ ( C ~ n ) ≅ H ∗ ( C F K − ( K ) ⊗ Q [ U ] Q [ U ] ( U n ) ) . H_*(C_n) \cong H_*(\widetilde{C}_n) \cong H_*\left(CFK^-(K) \mathbin{\otimes}_{\mathbb{Q}[U]} \frac{\mathbb{Q}[U]}{(U^n)}\right). H ∗ ( C n ) ≅ H ∗ ( C n ) ≅ H ∗ ( C F K − ( K ) ⊗ Q [ U ] ( U n ) Q [ U ] ) . using C / ( V ) ≃ C F K − ( K ) C/(V) \simeq CFK^-(K) C / ( V ) ≃ C F K − ( K ) Q [ U ] \mathbb{Q}[U] Q [ U ]
H ∗ ( C F K − ( K ) ⊗ Q [ U ] Q [ U ] ( U n ) ) ≅ [ H ∗ ( C F K − ( K ) ) ⊗ Q [ U ] Q [ U ] ( U n ) ] ⊕ Tor 1 Q [ U ] ( H ∗ ( C F K − ( K ) ) ; Q [ U ] ( U n ) ) . H_*\left(CFK^-(K) \mathbin{\otimes}_{\mathbb{Q}[U]} \frac{\mathbb{Q}[U]}{(U^n)}\right) \cong\left[H_*(CFK^-(K)) \mathbin{\otimes}_{\mathbb{Q}[U]} \frac{\mathbb{Q}[U]}{(U^n)}\right] \oplus\operatorname{Tor}^{\mathbb{Q}[U]}_1\left(H_*(CFK^-(K));\frac{\mathbb{Q}[U]}{(U^n)}\right). H ∗ ( C F K − ( K ) ⊗ Q [ U ] ( U n ) Q [ U ] ) ≅ [ H ∗ ( C F K − ( K )) ⊗ Q [ U ] ( U n ) Q [ U ] ] ⊕ Tor 1 Q [ U ] ( H ∗ ( C F K − ( K )) ; ( U n ) Q [ U ] ) . But
H F K − ( K ) = H ∗ ( C F K − ( K ) ) HFK^-(K) = H_*(CFK^-(K)) H F K − ( K ) = H ∗ ( C F K − ( K )) and Lemmata 4.13 and 4.12 yield
H F K n ( K ) ≅ H ∗ ( C F K − ( K ) ⊗ Q [ U ] Q [ U ] ( U n ) ) . HFK_n(K) \cong H_*\left(CFK^-(K) \mathbin{\otimes}_{\mathbb{Q}[U]} \frac{\mathbb{Q}[U]}{(U^n)}\right). H F K n ( K ) ≅ H ∗ ( C F K − ( K ) ⊗ Q [ U ] ( U n ) Q [ U ] ) . Piecing it all together gives the result. □
For L L L Theorem 4.14 is sufficient but not necessary. The following proposition computes H F K n ( K ) HFK_n(K) H F K n ( K ) L L L n ≥ 2 n \ge2 n ≥ 2 R = Q [ U ] ( U n ) R = \frac{\mathbb{Q}[U]}{(U^n)} R = ( U n ) Q [ U ]
Let n ≥ 2 n \ge2 n ≥ 2 C C C x 2 , x 4 , … , x m − 1 x_2,x_4,\ldots,x_{m-1} x 2 , x 4 , … , x m − 1 i i i h i = k i − 1 h_i=k_{i-1} h i = k i − 1 v i = k i v_i=k_i v i = k i x i x_i x i i i i h i ≥ n h_i \ge n h i ≥ n v i = 1 v_i=1 v i = 1 i i i r ( i ) r(i) r ( i ) r > i r>i r > i r = m + 1 r=m+1 r = m + 1 ( h r , v r ) ≠ ( n − 1 , 1 ) (h_r,v_r)\ne(n-1,1) ( h r , v r ) = ( n − 1 , 1 ) r ( i ) = m + 1 r(i)=m+1 r ( i ) = m + 1 h r ( i ) ≥ n h_{r(i)} \ge n h r ( i ) ≥ n N n ( C ) N_n(C) N n ( C ) L L L K K K N n ( K ) N_n(K) N n ( K ) N n ( C ) N_n(C) N n ( C ) C C C Theorem 4.11 . Then we have the following:
Proposition 4.15. As Q [ U ] \mathbb{Q}[U] Q [ U ]
H ∗ ( C n ) ≅ R ⊕ ⨁ i e v e n ( Q [ U ] ( U min ( h i , n ) ) ) 2 , H_*(C_n) \cong R \oplus\bigoplus_{i\ \mathrm{even}} \left(\frac{\mathbb{Q}[U]}{(U^{\min(h_i,n)})}\right)^2, H ∗ ( C n ) ≅ R ⊕ i even ⨁ ( ( U m i n ( h i , n ) ) Q [ U ] ) 2 , except that 2 N n ( C ) 2N_n(C) 2 N n ( C ) R R R Q [ U ] ( U n − 1 ) \frac{\mathbb{Q}[U]}{(U^{n-1})} ( U n − 1 ) Q [ U ]
dim Q H ∗ ( C n ) = n + 2 ∑ i e v e n min ( h i , n ) − 2 N n ( C ) , dim Q H ∗ ( C ~ n ) = n + 2 ∑ i e v e n min ( h i , n ) , \dim_{\mathbb{Q}} H_*(C_n)=n+2\sum_{i\ \mathrm{even}}\min(h_i,n)-2N_n(C), \qquad\dim_{\mathbb{Q}} H_*(\widetilde{C}_n)=n+2\sum_{i\ \mathrm{even}}\min(h_i,n), dim Q H ∗ ( C n ) = n + 2 i even ∑ min ( h i , n ) − 2 N n ( C ) , dim Q H ∗ ( C n ) = n + 2 i even ∑ min ( h i , n ) , and H ∗ ( C n ) ≅ H ∗ ( C ~ n ) H_*(C_n)\cong H_*(\widetilde{C}_n) H ∗ ( C n ) ≅ H ∗ ( C n ) N n ( C ) = 0 N_n(C)=0 N n ( C ) = 0 C n ≅ C ~ n C_n\cong\widetilde{C}_n C n ≅ C n R R R
Proof. As in the proof of Lemma 4.13 , C n C_n C n R R R x i x_i x i i i i x j x_j x j j j j ∂ n x i = U h i x i − 1 + ϵ i n U n − 1 x i + 1 \partial_nx_i=U^{h_i}x_{i-1}+\epsilon_i^nU^{n-1}x_{i+1} ∂ n x i = U h i x i − 1 + ϵ i n U n − 1 x i + 1 U h i = 0 U^{h_i}=0 U h i = 0 h i ≥ n h_i\ge n h i ≥ n D D D R R R C ~ n \widetilde{C}_n C n D ~ \widetilde{D} D U h i U^{h_i} U h i ( i − 1 , i ) (i-1,i) ( i − 1 , i )
We process i = 2 , 4 , … , m − 1 i=2,4,\ldots,m-1 i = 2 , 4 , … , m − 1 R R R i i i i − 1 i-1 i − 1 i i i c U n − 1 cU^{n-1} c U n − 1 c ∈ Q × c\in\mathbb{Q}^{\times} c ∈ Q × j j j
If h i ≤ n − 1 h_i\le n-1 h i ≤ n − 1 ( i − 1 , j ) (i-1,j) ( i − 1 , j ) x j x_j x j x j − c U n − 1 − h i x i x_j-cU^{n-1-h_i}x_i x j − c U n − 1 − h i x i ( i − 1 , j ) (i-1,j) ( i − 1 , j ) − c ϵ i n U 2 n − 2 − h i -c\epsilon_i^nU^{2n-2-h_i} − c ϵ i n U 2 n − 2 − h i ( i + 1 , j ) (i+1,j) ( i + 1 , j ) h i = n − 1 h_i=n-1 h i = n − 1 ϵ i = 1 \epsilon_i=1 ϵ i = 1 i + 1 i+1 i + 1 Lemma 4.13 , replace x i − 1 x_{i-1} x i − 1 x i − 1 + ϵ i n U n − 1 − h i x i + 1 x_{i-1}+\epsilon_i^nU^{n-1-h_i}x_{i+1} x i − 1 + ϵ i n U n − 1 − h i x i + 1 i − 1 i-1 i − 1 ( i + 1 , i ) (i+1,i) ( i + 1 , i )
If h i ≥ n h_i\ge n h i ≥ n i i i ϵ i n U n − 1 \epsilon_i^nU^{n-1} ϵ i n U n − 1 ( i + 1 , i ) (i+1,i) ( i + 1 , i ) i i i i + 1 i+1 i + 1 i − 1 i-1 i − 1 i − 1 i-1 i − 1
A new pending defect lands in row i + 1 i+1 i + 1 h i ≥ n h_i\ge n h i ≥ n ( h i , v i ) = ( n − 1 , 1 ) (h_i,v_i)=(n-1,1) ( h i , v i ) = ( n − 1 , 1 ) i i i i + 1 , i + 3 , … i+1,i+3,\ldots i + 1 , i + 3 , … ( h r , v r ) = ( n − 1 , 1 ) (h_r,v_r)=(n-1,1) ( h r , v r ) = ( n − 1 , 1 ) r > i r>i r > i ( h r , v r ) ≠ ( n − 1 , 1 ) (h_r,v_r)\ne(n-1,1) ( h r , v r ) = ( n − 1 , 1 ) h r ≤ n − 1 h_r\le n-1 h r ≤ n − 1 h r ≥ n h_r\ge n h r ≥ n r ≤ m − 1 r\le m-1 r ≤ m − 1 m m m Lemma 4.8 , the column operation is the zigzag term created when the arrow x i → U h i x i − 1 x_i\to U^{h_i}x_{i-1} x i → U h i x i − 1
All operations are invertible and R R R D ~ \widetilde{D} D c U n − 1 cU^{n-1} c U n − 1 D ~ \widetilde{D} D ( Q [ U ] ( U h i ) ) 2 \left(\frac{\mathbb{Q}[U]}{(U^{h_i})}\right)^2 ( ( U h i ) Q [ U ] ) 2 i i i h i ≤ n − 1 h_i\le n-1 h i ≤ n − 1 R → U h i R R\xrightarrow{U^{h_i}}R R U h i R ( Q [ U ] ( U n − 1 ) ) 2 \left(\frac{\mathbb{Q}[U]}{(U^{n-1})}\right)^2 ( ( U n − 1 ) Q [ U ] ) 2 R R R D ~ \widetilde{D} D R R R Q [ U ] ( U n − 1 ) \frac{\mathbb{Q}[U]}{(U^{n-1})} ( U n − 1 ) Q [ U ]
Remark 4.16. The staircase of the right form with steps k 1 , … , k m − 1 k_1,\ldots,k_{m-1} k 1 , … , k m − 1 k m − 1 , … , k 1 k_{m-1},\ldots,k_1 k m − 1 , … , k 1 C n C_n C n R R R Proposition 4.15 also computes H ∗ ( C n ) H_*(C_n) H ∗ ( C n ) H F K n HFK_n H F K n L L L
We illustrate Proposition 4.15 with two examples. The first is a staircase complex, not the master complex of a knot, chosen to show each of the three possibilities for a defect: it may survive, be carried to a later row, or be absorbed.
Example 4.17. Let C C C m = 9 m=9 m = 9
( k 1 , … , k 8 ) = ( 3 , 1 , 1 , 1 , 1 , 1 , 2 , 1 ) , (k_1,\ldots,k_8)=(3,1,1,1,1,1,2,1), ( k 1 , … , k 8 ) = ( 3 , 1 , 1 , 1 , 1 , 1 , 2 , 1 ) , so that the sources x 2 , x 4 , x 6 , x 8 x_2,x_4,x_6,x_8 x 2 , x 4 , x 6 , x 8
( h 2 , v 2 ) = ( 3 , 1 ) , ( h 4 , v 4 ) = ( 1 , 1 ) , ( h 6 , v 6 ) = ( 1 , 1 ) , ( h 8 , v 8 ) = ( 2 , 1 ) . (h_2,v_2)=(3,1),\qquad(h_4,v_4)=(1,1),\qquad(h_6,v_6)=(1,1),\qquad(h_8,v_8)=(2,1). ( h 2 , v 2 ) = ( 3 , 1 ) , ( h 4 , v 4 ) = ( 1 , 1 ) , ( h 6 , v 6 ) = ( 1 , 1 ) , ( h 8 , v 8 ) = ( 2 , 1 ) . The case n = 2 n=2 n = 2 Here R = Q [ U ] ( U 2 ) R=\frac{\mathbb{Q}[U]}{(U^2)} R = ( U 2 ) Q [ U ] C 2 C_2 C 2
∂ 2 x 2 = 2 U x 3 , ∂ 2 x 4 = U x 3 + 2 U x 5 , ∂ 2 x 6 = U x 5 + 2 U x 7 , ∂ 2 x 8 = 2 U x 9 , \partial_2x_2=2Ux_3,\qquad\partial_2x_4=Ux_3+2Ux_5,\qquad\partial_2x_6=Ux_5+2Ux_7,\qquad\partial_2x_8=2Ux_9, ∂ 2 x 2 = 2 U x 3 , ∂ 2 x 4 = U x 3 + 2 U x 5 , ∂ 2 x 6 = U x 5 + 2 U x 7 , ∂ 2 x 8 = 2 U x 9 , since U 3 = U 2 = 0 U^3=U^2=0 U 3 = U 2 = 0 x 2 , x 4 , x 6 , x 8 x_2,x_4,x_6,x_8 x 2 , x 4 , x 6 , x 8 x 1 , x 3 , x 5 , x 7 , x 9 x_1,x_3,x_5,x_7,x_9 x 1 , x 3 , x 5 , x 7 , x 9 D D D D ~ \widetilde{D} D Proposition 4.15 are
D = ( 0 0 0 0 2 U U 0 0 0 2 U U 0 0 0 2 U 0 0 0 0 2 U ) , D ~ = ( 0 0 0 0 0 U 0 0 0 0 U 0 0 0 0 0 0 0 0 0 ) . D=\begin{pmatrix}
0&0&0&0\\
2U&U&0&0\\
0&2U&U&0\\
0&0&2U&0\\
0&0&0&2U
\end{pmatrix},
\qquad
\widetilde{D}=\begin{pmatrix}
0&0&0&0\\
0&U&0&0\\
0&0&U&0\\
0&0&0&0\\
0&0&0&0
\end{pmatrix}. D = 0 2 U 0 0 0 0 U 2 U 0 0 0 0 U 2 U 0 0 0 0 0 2 U , D = 0 0 0 0 0 0 U 0 0 0 0 0 U 0 0 0 0 0 0 0 . Both x 2 x_2 x 2 x 8 x_8 x 8 Proposition 4.15 , the entry 2 U x 3 2Ux_3 2 U x 3 ∂ 2 x 2 \partial_2x_2 ∂ 2 x 2 x 3 x_3 x 3 x 4 x_4 x 4 h 4 = 1 ≤ n − 1 h_4=1\le n-1 h 4 = 1 ≤ n − 1 x 2 x_2 x 2 x 2 − 2 x 4 x_2-2x_4 x 2 − 2 x 4
∂ 2 ( x 2 − 2 x 4 ) = − 4 U x 5 . \partial_2(x_2-2x_4)=-4Ux_5. ∂ 2 ( x 2 − 2 x 4 ) = − 4 U x 5 . which is nonzero because ( h 4 , v 4 ) = ( n − 1 , 1 ) (h_4,v_4)=(n-1,1) ( h 4 , v 4 ) = ( n − 1 , 1 ) x 5 x_5 x 5 x 3 x_3 x 3 y 3 = x 3 + 2 x 5 y_3=x_3+2x_5 y 3 = x 3 + 2 x 5 ∂ 2 x 4 = U y 3 \partial_2x_4=Uy_3 ∂ 2 x 4 = U y 3 x 6 x_6 x 6 x 2 − 2 x 4 x_2-2x_4 x 2 − 2 x 4 x 2 ′ = x 2 − 2 x 4 + 4 x 6 x'_2=x_2-2x_4+4x_6 x 2 ′ = x 2 − 2 x 4 + 4 x 6 ∂ 2 x 2 ′ = 8 U x 7 \partial_2x'_2=8Ux_7 ∂ 2 x 2 ′ = 8 U x 7 y 5 = x 5 + 2 x 7 y_5=x_5+2x_7 y 5 = x 5 + 2 x 7 ∂ 2 x 6 = U y 5 \partial_2x_6=Uy_5 ∂ 2 x 6 = U y 5 x 8 x_8 x 8 h 8 = 2 ≥ n h_8=2\ge n h 8 = 2 ≥ n 8 U x 7 8Ux_7 8 U x 7 x 2 x_2 x 2 r ( 2 ) = 8 r(2)=8 r ( 2 ) = 8 x 8 x_8 x 8 r ( 8 ) = m + 1 r(8)=m+1 r ( 8 ) = m + 1 N 2 ( C ) = 2 N_2(C)=2 N 2 ( C ) = 2 { x 2 ′ , x 4 , x 6 , x 8 } \{x'_2,x_4,x_6,x_8\} { x 2 ′ , x 4 , x 6 , x 8 } { x 1 , y 3 , y 5 , x 7 , x 9 } \{x_1,y_3,y_5,x_7,x_9\} { x 1 , y 3 , y 5 , x 7 , x 9 } ∂ 2 \partial_2 ∂ 2
( 0 0 0 0 0 U 0 0 0 0 U 0 8 U 0 0 0 0 0 0 2 U ) , \begin{pmatrix}
0&0&0&0\\
0&U&0&0\\
0&0&U&0\\
8U&0&0&0\\
0&0&0&2U
\end{pmatrix}, 0 0 0 8 U 0 0 U 0 0 0 0 0 U 0 0 0 0 0 0 2 U , which is D ~ \widetilde{D} D 8 U 8U 8 U 2 U 2U 2 U Q ⊕ Q \mathbb{Q}\oplus\mathbb{Q} Q ⊕ Q x 1 x_1 x 1 R R R
H ∗ ( C 2 ) ≅ Q [ U ] ( U 2 ) ⊕ Q 8 , H ∗ ( C ~ 2 ) ≅ ( Q [ U ] ( U 2 ) ) 5 ⊕ Q 4 , H_*(C_2)\cong\frac{\mathbb{Q}[U]}{(U^2)}\oplus\mathbb{Q}^8,\qquad
H_*(\widetilde{C}_2)\cong\left(\frac{\mathbb{Q}[U]}{(U^2)}\right)^5\oplus\mathbb{Q}^4, H ∗ ( C 2 ) ≅ ( U 2 ) Q [ U ] ⊕ Q 8 , H ∗ ( C 2 ) ≅ ( ( U 2 ) Q [ U ] ) 5 ⊕ Q 4 , of dimensions 10 10 10 14 14 14 n + 2 ∑ min ( h i , n ) − 2 N 2 ( C ) = 2 + 2 ( 2 + 1 + 1 + 2 ) − 4 = 10 n+2\sum\min(h_i,n)-2N_2(C)=2+2(2+1+1+2)-4=10 n + 2 ∑ min ( h i , n ) − 2 N 2 ( C ) = 2 + 2 ( 2 + 1 + 1 + 2 ) − 4 = 10
The case n = 3 n=3 n = 3 Here R = Q [ U ] ( U 3 ) R=\frac{\mathbb{Q}[U]}{(U^3)} R = ( U 3 ) Q [ U ] ∂ 3 x 2 = 3 U 2 x 3 \partial_3x_2=3U^2x_3 ∂ 3 x 2 = 3 U 2 x 3 x 2 x_2 x 2 x 8 x_8 x 8 h 8 = 2 < n h_8=2<n h 8 = 2 < n x 4 x_4 x 4 x 2 x_2 x 2 x 2 − 3 U x 4 x_2-3Ux_4 x 2 − 3 U x 4
∂ 3 ( x 2 − 3 U x 4 ) = 3 U 2 x 3 − 3 U ( U x 3 + 3 U 2 x 5 ) = − 9 U 3 x 5 = 0. \partial_3(x_2-3Ux_4)=3U^2x_3-3U(Ux_3+3U^2x_5)=-9U^3x_5=0. ∂ 3 ( x 2 − 3 U x 4 ) = 3 U 2 x 3 − 3 U ( U x 3 + 3 U 2 x 5 ) = − 9 U 3 x 5 = 0. The defect is absorbed, because ( h 4 , v 4 ) = ( 1 , 1 ) ≠ ( n − 1 , 1 ) (h_4,v_4)=(1,1)\ne(n-1,1) ( h 4 , v 4 ) = ( 1 , 1 ) = ( n − 1 , 1 ) Proposition 4.15 , r ( 2 ) = 4 r(2)=4 r ( 2 ) = 4 h 4 ≤ n − 1 h_4\le n-1 h 4 ≤ n − 1 Lemma 4.13 , so N 3 ( C ) = 0 N_3(C)=0 N 3 ( C ) = 0 C 3 ≅ C ~ 3 C_3\cong\widetilde{C}_3 C 3 ≅ C 3
H ∗ ( C 3 ) ≅ H ∗ ( C ~ 3 ) ≅ ( Q [ U ] ( U 3 ) ) 3 ⊕ ( Q [ U ] ( U 2 ) ) 2 ⊕ Q 4 , H_*(C_3)\cong H_*(\widetilde{C}_3)\cong\left(\frac{\mathbb{Q}[U]}{(U^3)}\right)^3\oplus\left(\frac{\mathbb{Q}[U]}{(U^2)}\right)^2\oplus\mathbb{Q}^4, H ∗ ( C 3 ) ≅ H ∗ ( C 3 ) ≅ ( ( U 3 ) Q [ U ] ) 3 ⊕ ( ( U 2 ) Q [ U ] ) 2 ⊕ Q 4 , which is dimension 17 17 17
Example 4.18. The torus knot T 5 , 7 T_{5,7} T 5 , 7 12 12 12 Δ T 5 , 7 \Delta_{T_{5,7}} Δ T 5 , 7
( k 1 , … , k 16 ) = ( 1 , 4 , 1 , 1 , 1 , 2 , 1 , 1 , 1 , 2 , 1 , 1 , 1 , 4 , 1 ) . (k_1,\ldots,k_{16})=(1,4,1,1,1,2,1,1,1,2,1,1,1,4,1). ( k 1 , … , k 16 ) = ( 1 , 4 , 1 , 1 , 1 , 2 , 1 , 1 , 1 , 2 , 1 , 1 , 1 , 4 , 1 ) . So its complex is a staircase like that of Theorem 4.11 , which has m = 17 m=17 m = 17 x 2 , x 4 , … , x 16 x_2,x_4,\ldots,x_{16} x 2 , x 4 , … , x 16
( h i ) = ( 1 , 1 , 1 , 1 , 2 , 1 , 4 ) , ( v i ) = ( 4 , 1 , 2 , 1 , 1 , 1 , 1 ) . (h_i)=(1,1,1,1,2,1,4),\qquad(v_i)=(4,1,2,1,1,1,1). ( h i ) = ( 1 , 1 , 1 , 1 , 2 , 1 , 4 ) , ( v i ) = ( 4 , 1 , 2 , 1 , 1 , 1 , 1 ) . Take n = 2 n = 2 n = 2 x 12 x_{12} x 12 ( h 12 , v 12 ) = ( 2 , 1 ) (h_{12},v_{12}) = (2,1) ( h 12 , v 12 ) = ( 2 , 1 ) x 16 x_{16} x 16 ( h 16 , v 16 ) = ( 4 , 1 ) (h_{16},v_{16}) = (4,1) ( h 16 , v 16 ) = ( 4 , 1 ) x 12 x_{12} x 12 x 14 x_{14} x 14 ( h 14 , v 14 ) = ( 1 , 1 ) = ( n − 1 , 1 ) (h_{14},v_{14}) = (1,1) = (n-1,1) ( h 14 , v 14 ) = ( 1 , 1 ) = ( n − 1 , 1 ) x 16 x_{16} x 16 h 16 ≥ n h_{16} \ge n h 16 ≥ n x 16 x_{16} x 16 r ( 16 ) = m + 1 r(16) = m+1 r ( 16 ) = m + 1 N 2 ( T 5 , 7 ) = 2 N_2(T_{5,7}) = 2 N 2 ( T 5 , 7 ) = 2
HFK 2 ( T 5 , 7 ) ≅ Q [ U ] ( U 2 ) ⊕ Q 16 , \operatorname{HFK}_2(T_{5,7}) \cong\frac{\mathbb{Q}[U]}{(U^2)} \oplus\mathbb{Q}^{16}, HFK 2 ( T 5 , 7 ) ≅ ( U 2 ) Q [ U ] ⊕ Q 16 , which has dimension 18, while the right-hand side of Theorem 4.14 has dimension 22. For n = 3 n = 3 n = 3 x 16 x_{16} x 16 N 3 ( T 5 , 7 ) = 1 N_3(T_{5,7}) = 1 N 3 ( T 5 , 7 ) = 1 n = 4 n = 4 n = 4 N 4 ( T 5 , 7 ) = 1 N_4(T_{5,7}) = 1 N 4 ( T 5 , 7 ) = 1 n ≥ 5 n \ge5 n ≥ 5
Remark 4.19. For a thin knot K K K b b b HFK ^ − ( K ) ≅ Q [ U ] ⊕ Q ∣ τ ( K ) ∣ + 2 b \widehat{\operatorname{HFK}}^{-}(K) \cong\mathbb{Q}[U] \oplus\mathbb{Q}^{|\tau(K)|+2b} HFK − ( K ) ≅ Q [ U ] ⊕ Q ∣ τ ( K ) ∣ + 2 b rk HFK ^ ( K ) = 2 ∣ τ ( K ) ∣ + 1 + 4 b = det ( K ) \operatorname{rk}\widehat{\operatorname{HFK}}(K) = 2|\tau(K)| + 1 + 4b = \det(K) rk HFK ( K ) = 2∣ τ ( K ) ∣ + 1 + 4 b = det ( K ) HFK ^ ( K ) \widehat{\operatorname{HFK}}(K) HFK ( K ) δ \delta δ n ≥ 2 n \ge2 n ≥ 2
HFK n ( K ) ≅ Q [ U ] ( U n ) ⊕ Q det ( K ) − 1 2 ⊕ Q det ( K ) − 1 2 = Q [ U ] ( U n ) ⊕ Q det ( K ) − 1 , \operatorname{HFK}_n(K) \cong\frac{\mathbb{Q}[U]}{(U^n)} \oplus\mathbb{Q}^{\frac{\det(K)-1}{2}} \oplus\mathbb{Q}^{\frac{\det(K)-1}{2}} = \frac{\mathbb{Q}[U]}{(U^n)} \oplus\mathbb{Q}^{\det(K)-1}, HFK n ( K ) ≅ ( U n ) Q [ U ] ⊕ Q 2 d e t ( K ) − 1 ⊕ Q 2 d e t ( K ) − 1 = ( U n ) Q [ U ] ⊕ Q d e t ( K ) − 1 , which proves the second part of Corollary 1.5. For 2-bridge knots and n = 2 n = 2 n = 2 dim Q HFK 2 ( K ) = det ( K ) + 1 \dim_{\mathbb{Q}} \operatorname{HFK}_2(K) = \det(K) + 1 dim Q HFK 2 ( K ) = det ( K ) + 1 [1 Lemma 5.4] .
Proof of Corollary 1.6. For T p , q T_{p,q} T p , q S ⊂ Z ≥ 0 S \subset\mathbb{Z}_{\ge0} S ⊂ Z ≥ 0 p p p q q q
1 − t p q ( 1 − t p ) ( 1 − t q ) = ∑ s ∈ S t s \frac{1-t^{pq}}{(1-t^p)(1-t^q)} = \sum_{s \in S} t^s ( 1 − t p ) ( 1 − t q ) 1 − t pq = s ∈ S ∑ t s gives Δ T p , q ( t ) = ( 1 − t ) ∑ s ∈ S t s \Delta_{T_{p,q}}(t) = (1-t)\sum_{s \in S} t^s Δ T p , q ( t ) = ( 1 − t ) ∑ s ∈ S t s Δ T p , q \Delta_{T_{p,q}} Δ T p , q s ∈ S s \in S s ∈ S s − 1 ∉ S s-1 \notin S s − 1 ∈ / S s + 1 ∉ S s+1 \notin S s + 1 ∈ / S s ∈ S s \in S s ∈ S S S S S S S p − 1 p-1 p − 1 s ∈ S s \in S s ∈ S s + p ∈ S s+p \in S s + p ∈ S { 1 , … , p − 1 } \{1,\ldots,p-1\} { 1 , … , p − 1 } Δ \Delta Δ p − 1 p-1 p − 1
If n ≥ p n \ge p n ≥ p n − 1 n-1 n − 1 2 ≤ n ≤ p − 1 2 \le n \le p-1 2 ≤ n ≤ p − 1 Δ \Delta Δ 1 − t + t p 1-t+t^p 1 − t + t p p − 1 p-1 p − 1 x m − 1 x_{m-1} x m − 1 ( h m − 1 , v m − 1 ) = ( p − 1 , 1 ) (h_{m-1},v_{m-1}) = (p-1,1) ( h m − 1 , v m − 1 ) = ( p − 1 , 1 ) p − 1 ≥ n p-1 \ge n p − 1 ≥ n r ( m − 1 ) = m + 1 r(m-1) = m+1 r ( m − 1 ) = m + 1 N n ( T p , q ) ≥ 1 N_n(T_{p,q}) \ge1 N n ( T p , q ) ≥ 1 dim Q HFK n ( T p , q ) \dim_{\mathbb{Q}} \operatorname{HFK}_n(T_{p,q}) dim Q HFK n ( T p , q ) T 3 , 4 T_{3,4} T 3 , 4
The same argument shows more generally that, since the top gap of the Alexander polynomial of a nontrivial L L L [7 ] , the isomorphism of Theorem 4.14 fails for every nontrivial L L L n n n
References
DEPARTMENT OF MATHEMATICS, LOUISIANA STATE UNIVERSITY, LA 70803
Email address: trwells22@lsu.edu
