The conjugacy problem for groups of alternating prime tame links is polynomial-time

Abstract

Abstract. An alternating projection of a prime link can to used to produce a group presentation (of the link group under free product with the infinite cyclic group) with some useful peculiarities, including small cancellation conditions. In this presentation, conjugacy diagrams are shown to have the form of a tiling of squares in the Euclidean plane in one of a limited number of shapes. An argument based on the shape of the link projection is used to show that the tiling requires no more square tiles than a linear function of word length (with constant multiple based on the number of link crossings). It follows that the computation time for testing conjugacy of two group elements (previously known to be no worse than exponential) is bounded by a cubic polynomial. This bounds complexity in the original link group.

Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (I)

“…A trivial tangle is a pair (B 3 , t), where B 3 is a 3-ball and t is a union of two arcs properly embedded in B 3 which is parallel to a union of two mutually disjoint arcs in ∂B 3 . By a rational tangle, we mean a trivial tangle (B 3 , t) which is endowed with a homeomorphism from (∂B 3 , ∂t) to (S 2 , P ).…”

“…It has been proved by Weinbaum [16] and Appel and Schupp [1] that the word and conjugacy problems for prime alternating link groups are solvable, by using small cancellation theory (see also [3] and references in it). Moreover, it was also shown by Sela [15] and Préaux [12] that the word and conjugacy problems for any link group are solvable.…”

“…2,3,4). Then the following hold, where v ib and v ie are nonempty initial and terminal subwords ofv i with |v ib |, |v ie | ≤ |v i | − 1, respectively.…”

Non-peripheral ideal decompositions of alternating knots

“…Notice that in the proof for the non-peripherality of short arcs we actually solved the conjugacy problem. We could also have shown that these elements were nonperipheral by using Johnsgard's solution to the conjugacy problem [Joh97]: using Johnsgard's algorithm, the fact that the peripheral complex contains the geodesic completion of l m , and the periodicity of the peripheral complex, it is straight-forward to show that a geodesic word in the augmented Dehn presentation is conjugate to a peripheral element l a m b if and only if it embeds as a geodesic path from (0, k) to (an + k − b, an + b), for some integer k. It is easy to see that two words in (5) and ( 6) are not of this form. We can use a similar argument for Wirtinger loops.…”

Simple loops on 2-bridge spheres in 2-bridge link complements

“…The key tool for solving the question is small cancellation theory, applied to two-generator and one-relator presentations of 2-bridge link groups. We note that it has been proved by Weinbaum [32] and Appel and Schupp [5] that the word and conjugacy problems for prime alternating link groups are solvable, by using small cancellation theory (see also [10] and references in it). Moreover, it was shown by Sela [24] and Préaux [21] that the word and conjugacy problems for any link group are solvable.…”