Polynomial-Time Algorithms for Links

Every link in the $3$-sphere admits a generalised $T$-link presentation. In this paper, we give an explicit polynomial-time algorithm which, given a signed Artin braid word and the number of strands, computes such a presentation. Combining this construction with Yamada–Vogel braiding gives a polynomial-time conversion from arbitrary oriented combinatorial link diagrams to generalised $T$-link parameters. We prove polynomial bounds on the number of strands and crossings in the resulting braid. As an application, we prove that a link property or a link invariant is computable in polynomial time from diagrams if and only if the corresponding problem is solvable in polynomial time from generalised $T$-link parameters, with parameter input measured by the total number of strands and crossings in the corresponding braid diagram. Whenever a classification criterion or an explicit invariant formula is available in these parameters, it can be applied to the original link by substituting the parameters obtained from the diagram. A polynomial-time test or evaluation then gives a polynomial-time procedure on diagrams. We also establish the corresponding transfer for approximating real-valued invariants to prescribed accuracy.