Computing with knot quandles

2018 ◽  
Vol 27 (14) ◽  
pp. 1850074
Author(s):  
Graham Ellis ◽  
Cédric Fragnaud
Keyword(s):  
Abelian Groups ◽  
Finitely Generated ◽  
Computational Algebra ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Knot Invariant ◽  
Number Formula ◽  
Finitely Presented

The number [Formula: see text] of colorings of a knot [Formula: see text] by a finite quandle [Formula: see text] has been used in the literature to distinguish between knot types. In this paper, we suggest a refinement [Formula: see text] to this knot invariant involving any computable functor [Formula: see text] from finitely presented groups to finitely generated abelian groups. We are mainly interested in the functor [Formula: see text] that sends each finitely presented group [Formula: see text] to its abelianization [Formula: see text]. We describe algorithms needed for computing the refined invariant and illustrate implementations that have been made available as part of the HAP package for the GAP system for computational algebra. We use these implementations to investigate the performance of the refined invariant on prime knots with [Formula: see text] crossings.

On groups and 3-manifolds with weak dimension ≤ 1

1974 ◽  
Vol 75 (1) ◽  
pp. 33-35
Author(s):  
David E. Galewski
Keyword(s):  
Fundamental Group ◽  
Finitely Generated ◽  
Finite Generation ◽  
Connected Sum ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Weak Dimension ◽  
Finitely Presented

0. Introduction. A group π has weak dimension (wd) ≤ n (see Cartan and Ellen-berg (2)) if Hk(π, A) = 0 for all right Z(π)-modules A and all k > n. We say that the weak dimension of a manifold M is ≤ n if wd (πl(M))≤ n. In section 1 we show that open, orientable, irreducible 3-manifolds have wd ≤ 1 if and only if they are the monotone on of 1-handle bodies. In his celebrated theorem (10), Stallings proves that finitely presented groups of cohomological dimensions ≤ 1 are free. In section 2 we prove that if π is a finitely presented group which is the fundamental group of any orientable 3-manifold with wd ≤ 1 then π is free. We also give an example to show that the finite generation of π is necessary. (Swan (11) removes the finitely presented hypothesis from Stalling's theorem.) Finally, in section 3 we generalize a theorem of McMillan (5) and prove that if M is an open, orientable, irreducible 3-manifold with finitely generated fundamental group, then M is stably (taking the product with n ≥ 1 copies of ℝ) a connected sum along the boundary of trivial (n+2)-disc Sl bundles.

scholarly journals INDICABLE GROUPS AND ENDOMORPHIC PRESENTATIONS

2011 ◽  
Vol 54 (2) ◽  
pp. 335-344
Author(s):  
MUSTAFA GÖKHAN BENLI
Keyword(s):  
Normal Subgroup ◽  
Semidirect Product ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Free Semigroups ◽  
Finitely Presented ◽  
Indicable Group

AbstractIn this paper we look at presentations of subgroups of finitely presented groups with infinite cyclic quotients. We prove that if H is a finitely generated normal subgroup of a finitely presented group G with G/H cyclic, then H has ascending finite endomorphic presentation. It follows that any finitely presented indicable group without free semigroups has the structure of a semidirect product H ⋊ ℤ, where H has finite ascending endomorphic presentation.

Decision problems concerning S-arithmetic groups

1985 ◽  
Vol 50 (3) ◽  
pp. 743-772 ◽  
Author(s):  
Fritz Grunewald ◽  
Daniel Segal
Keyword(s):  
Additive Group ◽  
Nilpotent Groups ◽  
Isomorphism Problem ◽  
Algorithmic Problem ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Finite Dimensional ◽  
Associative Rings ◽  
Finitely Presented ◽  
Torsion Free

This paper is a continuation of our previous work in [12]. The results, and some applications, have been described in the announcement [13]; it may be useful to discuss here, a little more fully, the nature and purpose of this work.We are concerned basically with three kinds of algorithmic problem: (1) isomorphism problems, (2) “orbit problems”, and (3) “effective generation”.(1) Isomorphism problems. Here we have a class of algebraic objects of some kind, and ask: is there a uniform algorithm for deciding whether two arbitrary members of are isomorphic? In most cases, the answer is no: no such algorithm exists. Indeed this has been one of the most notable applications of methods of mathematical logic in algebra (see [26, Chapter IV, §4] for the case where is the class of all finitely presented groups). It turns out, however, that when consists of objects which are in a certain sense “finite-dimensional”, then the isomorphism problem is indeed algorithmically soluble. We gave such algorithms in [12] for the following cases: = {finitely generated nilpotent groups}; = {(not necessarily associative) rings whose additive group is finitely generated}; = {finitely Z-generated modules over a fixed finitely generated ring}.Combining the methods of [12] with his own earlier work, Sarkisian has obtained analogous results with the integers replaced by the rationals: in [20] and [21] he solves the isomorphism problem for radicable torsion-free nilpotent groups of finite rank and for finite-dimensional Q-algebras.

Separating Classes of Groups by First-Order Sentences

2003 ◽  
Vol 13 (03) ◽  
pp. 287-302 ◽  
Author(s):  
André Nies
Keyword(s):  
Nilpotent Group ◽  
Word Problem ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
First Order ◽  
Solvable Word Problem ◽  
Rank 2 ◽  
Finitely Presented ◽  
Solvable Word

For various proper inclusions of classes of groups [Formula: see text], we obtain a group [Formula: see text] and a first-order sentence φ such that H⊨φ but no G∈ C satisfies φ. The classes we consider include the finite, finitely presented, finitely generated with and without solvable word problem, and all countable groups. For one separation, we give an example of a f.g. group, namely ℤp ≀ ℤ for some prime p, which is the only f.g. group satisfying an appropriate first-order sentence. A further example of such a group, the free step-2 nilpotent group of rank 2, is used to show that true arithmetic Th(ℕ,+,×) can be interpreted in the theory of the class of finitely presented groups and other classes of f.g. groups.

scholarly journals Embeddings into groups with only a few defining relations

1974 ◽  
Vol 18 (1) ◽  
pp. 1-7 ◽  
Author(s):  
W. W. Boone ◽  
D. J. Collins
Keyword(s):  
Word Problem ◽  
Finitely Presented Groups ◽  
Fixed Group ◽  
Defining Relations ◽  
Finitely Presented Group ◽  
One Relator Groups ◽  
Finitely Presented ◽  
Unsolvable Word Problem ◽  
Trivial Consequence

It is a trivial consequence of Magnus' solution to the word problem for one-relator groups [9] and the existence of finitely presented groups with unsolvable word problem [4] that not every finitely presented group can be embedded in a one-relator group. We modify a construction of Aanderaa [1] to show that any finitely presented group can be embedded in a group with twenty-six defining relations. It then follows from the well-known theorem of Higman [7] that there is a fixed group with twenty-six defining relations in which every recursively presented group is embedded.

Reflections on some aspects of infinite groups

2014 ◽  
Vol 6 (2) ◽  
Author(s):  
Benjamin Fine ◽  
Anthony Gaglione ◽  
Gerhard Rosenberger ◽  
Dennis Spellman
Keyword(s):  
Finitely Generated ◽  
Infinite Groups ◽  
Limit Groups ◽  
Finitely Presented Groups ◽  
The Relationship ◽  
Finitely Presented

AbstractIn this paper we survey and reflect upon several aspects of the theory of infinite finitely generated and finitely presented groups that were originally motivated by work of Gilbert Baumslag. All but the last of the topics we have chosen are all related in one way or another to the theory of limit groups and the solution of the Tarski problems. These include the residually free and fully residually free properties and the big powers condition; Baumslag doubles and extensions of centralizers; residually-𝒳 groups and extensions of results of Benjamin Baumslag and finally the relationship between CT and CSA groups.

scholarly journals Direct products and properly 3-realisable groups

2004 ◽  
Vol 70 (2) ◽  
pp. 199-205 ◽  
Author(s):  
Manuel Cárdenas ◽  
Francisco F. Lasheras ◽  
Ranja Roy
Keyword(s):  
Homotopy Type ◽  
Universal Cover ◽  
Hyperbolic Groups ◽  
Direct Products ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Manifold With Boundary ◽  
Proper Homotopy ◽  
Finitely Presented

In this paper, we show that the direct of infinite finitely presented groups is always properly 3-realisable. We also show that classical hyperbolic groups are properly 3-realisable. We recall that a finitely presented group G is said to be properly 3-realisable if there exists a compact 2-polyhedron K with π1 (K) ≅ G and whose universal cover K̃ has the proper homotopy type of a (p.1.) 3-manifold with boundary. The question whether or not every finitely presented is properly 3-realisable remains open.

A FINITELY PRESENTED GROUP WITH ALMOST SOLVABLE CONJUGACY PROBLEM

2009 ◽  
Vol 02 (04) ◽  
pp. 611-635 ◽  
Author(s):  
K. Kalorkoti
Keyword(s):  
Conjugacy Problem ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Recursively Enumerable ◽  
Computation Step ◽  
Solvable Word Problem ◽  
Finitely Presented ◽  
Solvable Word ◽  
Extended Notion ◽  
Ordered Pairs

The algorithmic unsolvability of the conjugacy problem for finitely presented groups was demonstrated by Novikov in the early 1950s. Various simplifications and alternative proofs were found by later researchers and further questions raised. Recent work by Borovik, Myasnikov and Remeslennikov has considered the question of what proportion of the number of elements of a group (obtained by standard constructions) falls into the realm of unsolvability. In this paper we provide a straightforward construction, as a Britton tower, of a finitely presented group with solvable word problem but unsolvable conjugacy problem of any r.e. (recursively enumerable) Turing degree a. The question of whether two elements are conjugate is bounded truth-table reducible to the question of whether the elements are both conjugate to a single generator of the group. We also define computable normal forms, based on the method of Bokut', that are suitable for the conjugacy problem. We consider (ordered) pairs of normal words U, V for the conjugacy problem whose lengths add to l and show that the proportion of such pairs for which conjugacy is undecidable (in the case a ≠ 0) is strictly less than l2/(2λ - 1)l where λ > 4. The construction is based on modular machines, introduced by Aanderaa and Cohen. For the purposes of this construction it was helpful to extend the notion of configuration to include pairs of m-adic integers. The notion of computation step was also extended and is referred to as s-fold computation where s ∈ ℤ (the usual notion coresponds to s = 1). If gcd (m, s) = 1 then determinism is preserved, i.e., if the modular machine is deterministic then it remains so under the extended notion. Furthermore there is a simple correspondence between s-fold and standard computation in this case. Otherwise computation is non-deterministic and there does not seem to be any straightforward correspondence between s-fold and standard computation.

scholarly journals A periodicity theorem for acylindrically hyperbolic groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Oleg Bogopolski
Keyword(s):  
Free Groups ◽  
Hyperbolic Groups ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Periodicity Theorem ◽  
Finitely Presented

AbstractWe generalize a well-known periodicity lemma from the case of free groups to the case of acylindrically hyperbolic groups. This generalization has been used to describe solutions of certain equations in acylindrically hyperbolic groups and to characterize verbally closed finitely generated acylindrically hyperbolic subgroups of finitely presented groups.

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