Topological Insights and Combinatorial Frameworks for Twist Knot Complements

The complement of a knot in $\mathbb{S}^3$ is a fundamental object in $3$-manifold topology and serves as a primary building block in the study of geometric structures on manifolds. For a large class of knots, these complements admit complete hyperbolic metrics, motivating the development of various frameworks for their classification and analysis. A twist knot is formed by applying a series of half-twists to a closed loop and linking the ends. In this article, we present explicit topological and combinatorial descriptions of all twist knot complements via the construction of suitable cell structures. Our method produces a closed $3$-dimensional pseudomanifold with a single vertex, arising from prescribed face pairings of a faceted $3$-ball. Removing the vertex yields the complement of the corresponding twist knot. The construction naturally induces ideal triangulations of these knot complements, several of which are shown to be facet-minimal. Beyond providing concrete models, our approach offers new perspectives on the topology of knot complements and leads to some open research problems.