When can a link be obtained from another using crossing exchanges and smoothings?

Abstract

Let L be a fixed link. Given a link diagram D, is there a sequence of crossing exchanges and smoothings on D that yields a diagram of L? We approach this problem from the computational complexity point of view. It follows from work by Endo, Itoh, and Taniyama that if L is a prime link with crossing number at most 5, then there is an algorithm that answers this question in polynomial time. We show that the same holds for all torus links T2,m and all twist knots.

Well-quasi-order of plane minors and an application to link diagrams

“…It was conjectured in [42] that for each fixed link L, there is a polynomial-time algorithm that solves ; L. Using Theorems 1.1 and 1.3, we settle this conjecture in the affirmative:…”

“…There is an extensive body of literature dealing with the question of when two links (or link diagrams or projections) are related under some set of local operations [1, 2, 8, 14, 15, 17-19, 22, 24-35, 37-39, 41, 44, 48-50]. We refer the reader to [42] for a detailed discussion on these references.…”

“…In [42] it was proved that if L is a torus link T 2,m or a twist knot, then there is a polynomial-time algorithm that solves ; L. That is, there is a polynomial p(n) and an algorithm A L that takes as input any diagram D with n crossings, and answers whether or not D ; L in at most p(n) steps. It was also pointed out in that paper that it follows from results in [14] that if L is any prime link with crossing number at most 5, then there is also a polynomial-time algorithm that solves ; L.…”

“…In the final section of [42] the following open question was included, remarking that if it were shown that if for every fixed link L there is a polynomial-time algorithm that solves ; L, then this would be the next natural problem to tackle. Since in this paper we have precisely settled the issue for each fixed link L, it seems worth reiterating this open problem.…”