Twisted conjugacy in dihedral Artin groups I: Torus Knot groups

Several distinct Garside monoids having torus knot groups as groups of fractions are known. For [Formula: see text] two coprime integers, we introduce a new Garside monoid [Formula: see text] having as Garside group the [Formula: see text]-torus knot group, thereby generalizing to all torus knot groups a construction that we previously gave for the [Formula: see text]-torus knot group. As a byproduct, we obtain new Garside structures for the braid groups of a few exceptional complex reflection groups of rank two. Analogous Garside structures are also constructed for a few additional braid groups of exceptional complex reflection groups of rank two which are not isomorphic to torus knot groups, namely, for [Formula: see text] and for dihedral Artin groups of even type.

In the first part of this paper, we present general results concerning the colorability of torus knots using conjugation quandles over any abstract group. Subsequently, we offer a numerical characterization for the colorability of torus knots using conjugation quandles over some particular groups: the matrix groups [Formula: see text] and [Formula: see text], the dihedral group, and the symmetric group.

Conjugacy problem for braid groups and Garside groups

The conjugacy problem for a finitely generated group G is the two-variable problem of deciding for an arbitrary pair [Formula: see text] of elements of G, whether or not u is conjugate to v in G. We construct examples of finitely generated, computably presented groups such that for every element [Formula: see text] of G, the problem of deciding if an arbitrary element is conjugate to [Formula: see text] is decidable in quadratic time but the worst-case complexity of the global conjugacy problem is arbitrary: it can be any c.e. Turing degree, can exactly mirror the Time Hierarchy Theorem, or can be [Formula: see text]-complete. Our groups also have the property that the conjugacy problem is generically linear time: that is, there is a linear time partial algorithm for the conjugacy problem whose domain has density 1, so hard instances are very rare. We also consider the complexity relationship of the “half-conjugacy” problem to the conjugacy problem. In the last section we discuss the extreme opposite situation: groups with algorithmically finite conjugation.

In this note we solve the twisted conjugacy problem for braid groups, i.e. we propose an algorithm which, given two braids $u,v\in B_n$ and an automorphism $\phi \in Aut (B_n)$, decides whether $v=(\phi (x))^{-1}ux$ for some $x\in B_n$. As a corollary, we deduce that each group of the form $B_n \rtimes H$, a semidirect product of the braid group $B_n$ by a torsion-free hyperbolic group $H$, has solvable conjugacy problem.

Given a short exact sequence of groups with certain conditions, 1 → F → G → H → 1 1\rightarrow F\rightarrow G\rightarrow H\rightarrow 1 , we prove that G G has solvable conjugacy problem if and only if the corresponding action subgroup A ⩽ A u t ( F ) A\leqslant Aut(F) is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form Z 2 ⋊ F m \mathbb {Z}^2\rtimes F_m , F 2 ⋊ F m F_2\rtimes F_m , F n ⋊ Z F_n \rtimes \mathbb {Z} , and Z n ⋊ A F m \mathbb {Z}^n \rtimes _A F_m with virtually solvable action group A ⩽ G L n ( Z ) A\leqslant GL_n(\mathbb {Z}) . Also, we give an easy way of constructing groups of the form Z 4 ⋊ F n \mathbb {Z}^4\rtimes F_n and F 3 ⋊ F n F_3\rtimes F_n with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and we give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in A u t ( F 2 ) Aut(F_2) is given.

A 1-bridge torus knot in a 3-manifold of genus ≤ 1 is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form and the Conway's normal form for 2-bridge knots. For a given Schubert's normal form we give algorithms to determine the number of components and to compute the fundamental group of the complement when the normal form determines a knot. We also give a description of the double branched cover of an ambient 3-manifold branched along a 1-bridge torus knot by using its Conway's normal form and obtain an explicit formula for the first homology of the double cover.

Amenability of Schreier graphs and strongly generic algorithms for the conjugacy problem

In various occasions the conjugacy problem in finitely generated amalgamated products and HNN extensions can be decided efficiently for elements which cannot be conjugated into the base groups. This observation asks for a bound on how many such elements there are. Such bounds can be derived using the theory of amenable graphs:In this work we examine Schreier graphs of amalgamated products and HNN extensions. For an amalgamated product G=H*AK with [H:A]≥[K:A] ≥2, the Schreier graph with respect to H or K turns out to be non-amenable if and only if [H:A]≥ 3. Moreover, for an HNN extension of the form G = , we show that the Schreier graph of G with respect to the subgroup H is non-amenable if and only if A ≠H ≠ φ(A).As application of these characterizations we show that under certain conditions the conjugacy problem in fundamental groups of finite graphs of groups with free abelian vertex groups can be solved in polynomial time on a strongly generic set. Furthermore, the conjugacy problem in groups with more than one end can be solved with a strongly generic algorithm which has essentially the same time complexity as the word problem. These are rather striking results as the word problem might be easy, but the conjugacy problem might be even undecidable. Finally, our results yield another proof that the set where the conjugacy problem of the Baumslag group G1,2 is decidable in polynomial time is also strongly generic.

We show that the number of homomorphisms from a knot group to a finite group G cannot be a Vassiliev invariant, unless it is constant on the set of (2, 2p+1) torus knots. In several cases, such as when G is a dihedral or symmetric group, this implies that the number of homomorphisms is not a Vassiliev invariant.

We prove that the conjugacy problem in the first Grigorchuk group [Formula: see text] can be solved in linear time. Furthermore, the problem to decide if a list of elements [Formula: see text] contains a pair of conjugate elements can be solved in linear time. We also show that a conjugator for a pair of conjugate element [Formula: see text] can be found in polynomial time.

We solve the twisted conjugacy problem on Thompson’s group F. We also exhibit orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and Aut+(F) are orbit decidable provided a certain conjecture on Thompson’s group T is true. By using general criteria introduced by Bogopolski, Martino and Ventura in [5], we construct a family of free extensions of F where the conjugacy problem is unsolvable. As a byproduct of our techniques, we give a new proof of a result of Bleak–Fel’shtyn–Goncalves in [4] showing that F has property R∞, and which can be extended to show that Thompson’s group T also has property R∞.

We establish a connection between the generalized conjugacy problem for a G-by- group, , and two algorithmic problems for G: the generalized Brinkmann’s conjugacy problem, GBrCP(G), and the generalized twisted conjugacy problem, GTCP(G). We explore this connection for generalizations of different kinds: relative to finitely generated subgroups, to their cosets, or to recognizable, rational, context-free or algebraic subsets of the group. Using this result, we are able to prove that GBrCP(G) is decidable (with respect to cosets) when G is a virtually polycyclic group, which implies in particular that the generalized Brinkmann’s equality problem, GBrP(G), is decidable if G is a finitely generated abelian group. Finally, we prove that if G is a finitely generated virtually free group, then the simple versions of Brinkmann’s equality problem and of the twisted conjugacy problem, BrP(G) and TCP(G), are decidable.

We exhibit a certain infinite family of three-stranded quasi-alternating pretzel knots which are counterexamples to Lobb's conjecture that the sl_3-knot concordance invariant s_3 (suitably normalised) should be equal to the Rasmussen invariant s_2. For this family, |s_3| < |s_2|. However, we also find other knots for which |s_3| > |s_2|. The main tool is an implementation of Morrison and Nieh's algorithm to calculate Khovanov's sl_3-foam link homology. Our C++-program is fast enough to calculate the integral homology of e.g. the (6,5)-torus knot in six minutes. Furthermore, we propose a potential improvement of the algorithm by gluing sub-tangles in a more flexible way.

There is an important class of finitely presented groups known as automatic groups, in which many computations can be undertaken efficiently. In particular, there is a normal form for group elements which is the language of a finite state automation, and arbitrary words in the group generators can be reduced to normal form in quadratic time. The normal form can also be used to enumerate group elements, and to compute the (rational) growth function of the group. The theory of automatic groups is developed [2], and the algorithms are described in more detail in [4]. Implementations of the algorithms are available in a software package of the author ([3]).The word-hyperbolic groups form an important subclass of the automatic groups. They are defined by the property that geodesic triangles in the Cayley graph of the group are uniformly thin. In other words, there is a constant e, such a that any point on one of the sides of such a triangle is within distance e of the union of the other two sides of the triangle. Small cancellation groups and groups acting discretely and cocompactly on hyperbolic space are examples of word-hyperbolic groups. Their theory is developed in [1].The word problem in a word-hyperbolic group can be solved by a Dehn algorithm in linear time. Furthermore, Epstein has shown that the conjugacy problem can be solved in time O(n ln n). However, both of these algorithms require a prior knowledge of the constant e, and it does not seem to be easy to compute this constant in general.In practice, the quadratic time algorithm for the word problem coming from the automatic groups procedures performs very effectively, and it would be useful to have alternatives for the conjugacy problem.The automatic groups machinery does provide such an algorithm (which works in the the intermediate class of biautomatic groups), but that unfortunately is exponential.Alternative algorithms for conjugacy testing of elements and subgroups are currently being developed and implemented by a research student of mine, Joe Marshall. They have the minor disadvanatage that their basic versions work only in torsion-free groups, but adaptations to the more general case seem possible. They include also a test for malnormality of a subgroup H of G, which means that g-1Hg∩H = 1 for all g ∈ G\H. This test is important in topological applications.They make essential use of the boundary of the group. The points on the boundary can be defined as the equivalence classes of infinite geodesic rays starting from the base point of the Cayley graph, where two such rays are equivalent if they remain a bounded distance apart. The precise bound involved here turns out to be another universal constant d of the Cayley graph, but unlike e it can be computed in practice relatively easily. Roughly speaking, a certain linear function of d provides us with a bound on the length of the elements g that we need to consider as possible conjugating elements when testing for conjugacy or malnormality. To make the algorithm practical, we need to restrict this set of potential conjugators considerably more than this, and Marshall has come up with various tricks for doing this.