scholarly journals Generalized Torsion in Knot Groups

2016 ◽  
Vol 59 (01) ◽  
pp. 182-189 ◽  
Author(s):  
Geoff Naylor ◽  
Dale Rolfsen
Keyword(s):  
Fundamental Group ◽  
Torus Knots ◽  
Torsion Element ◽  
Nonidentity Element ◽  
Group A

Abstract In a group, a nonidentity element is called a generalized torsion element if some product of its conjugates equals the identity. We show that for many classical knots one can ûnd generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the (hyperbolic) knot 52, and algebraic knots in the sense of Milnor.

scholarly journals Parameterizations of 1-Bridge Torus Knots

2003 ◽  
Vol 12 (04) ◽  
pp. 463-491 ◽  
Author(s):  
Doo Ho Choi ◽  
Ki Hyoung Ko
Keyword(s):  
Normal Form ◽  
Fundamental Group ◽  
Explicit Formula ◽  
Normal Forms ◽  
Double Cover ◽  
Torus Knots ◽  
Number Of Components ◽  
Torus Knot ◽  
Branched Cover ◽  
The Family

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.

scholarly journals PD4-complexes with fundamental group a PD2-group

2004 ◽  
Vol 142 (1-3) ◽  
pp. 49-60 ◽  
Author(s):  
Jonathan A Hillman
Keyword(s):  
Fundamental Group ◽  
Group A

scholarly journals Distinguishing the generalized knot groups of square and granny knot analogues

2019 ◽  
Vol 28 (05) ◽  
pp. 1950035
Author(s):  
Howida Al Fran ◽  
Christopher Tuffley
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Torus Knots ◽  
Connected Sums ◽  
The Difference ◽  
A Finite Group

Given a knot [Formula: see text], we may construct a group [Formula: see text] from the fundamental group of [Formula: see text] by adjoining an [Formula: see text]th root of the meridian that commutes with the corresponding longitude. For [Formula: see text] these “generalized knot groups” determine [Formula: see text] up to reflection. The second author has shown that for [Formula: see text], the generalized knot groups of the square and granny knots can be distinguished by counting homomorphisms into a suitably chosen finite group. We extend this result to certain generalized knot groups of square and granny knot analogues [Formula: see text], [Formula: see text], constructed as connected sums of [Formula: see text]-torus knots of opposite or identical chiralities. More precisely, for coprime [Formula: see text] and [Formula: see text] satisfying a coprimality condition with [Formula: see text] and [Formula: see text], we construct an explicit finite group [Formula: see text] (depending on [Formula: see text], [Formula: see text] and [Formula: see text]) such that [Formula: see text] and [Formula: see text] can be distinguished by counting homomorphisms into [Formula: see text]. The coprimality condition includes all [Formula: see text] coprime to [Formula: see text]. The result shows that the difference between these two groups can be detected using a finite group.

The Σ1-invariant for Artin groups of circuit rank 1

2015 ◽  
Vol 27 (5) ◽  
Author(s):  
Kisnney Almeida ◽  
Dessislava Kochloukova
Keyword(s):  
Fundamental Group ◽  
Free Group ◽  
Connected Graph ◽  
Artin Group ◽  
Artin Groups ◽  
Group A

AbstractWe define an Artin group of circuit rank 1 as an Artin group based on a connected graph with fundamental group a free group of rank 1. For an Artin group

scholarly journals Distinguishing Maps II: General Case

10.37236/3410 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Thomas W. Tucker
Keyword(s):  
Automorphism Group ◽  
Complete Classification ◽  
Distinguishing Number ◽  
Nonidentity Element ◽  
Vertex Set ◽  
Group A

A group $A$ acting faithfully on a set $X$ has  distinguishing number $k$, written $D(A,X)=k$, if there is a coloring of the elements of $X$ with $k$ colors such that no nonidentity element of $A$ is color-preserving, and no such coloring with fewer than $k$ colors exists.  Given a map $M$ with vertex set $V$ and automorphism group $Aut(M)$, let $D(M)=D(Aut(M),V)$. If $M$ is orientable, let $D^+(M)=D(Aut^+(M),V)$, where $Aut^+(M)$ is the group of orientation-preserving automorphisms.   In a previous paper, the author showed there are four maps $M$ with $D^+(M)>2$.  In this paper,  a complete classification is given for the graphs underlying maps with $D(M)>2$. There are $31$ such graphs, $22$ having no vertices of valence $1$ or $2$, and all have at most $10$ vertices.

A FORMULA FOR THE A-POLYNOMIAL OF TWIST KNOTS

2004 ◽  
Vol 13 (02) ◽  
pp. 193-209 ◽  
Author(s):  
JIM HOSTE ◽  
PATRICK D. SHANAHAN
Keyword(s):  
Fundamental Group ◽  
Simple Form ◽  
Infinite Family ◽  
Newton Polygons ◽  
Torus Knots ◽  
Single Relation

The fundamental group of a 2-bridge knot has a particularly nice presentation, having only two generators and a single relation. For certain families of 2-bridge knots, such as the torus knots, or the twist knots, the relation takes on an especially simple form. Exploiting this form, we derive a formula for the A-polynomial of twist knots. Our methods extend to at least one other infinite family of (non-torus) 2-bridge knots. Using these formulae we determine the associated Newton polygons. We further prove that the A-polynomials of twist knots are irreducible.

scholarly journals Non-left-orderable surgeries on 1-bridge braids

2020 ◽  
Vol 29 (12) ◽  
pp. 2050086
Author(s):  
Shiyu Liang
Keyword(s):  
Fundamental Group ◽  
Rational Homology ◽  
Torus Knots ◽  
Large Families

Boyer, Gordon and Watson have conjectured that an irreducible rational homology [Formula: see text]-sphere is an L-space if and only if its fundamental group is not left-orderable. Since Dehn surgeries on knots in [Formula: see text] can produce large families of L-spaces, it is natural to examine the conjecture on these [Formula: see text]-manifolds. Greene, Lewallen and Vafaee have proved that all [Formula: see text]-bridge braids are L-space knots. In this paper, we consider three families of [Formula: see text]-bridge braids. First we calculate the knot groups and peripheral subgroups. We then verify the conjecture on the three cases by applying the criterion developed by Christianson, Goluboff, Hamann and Varadaraj, when they verified the same conjecture for certain twisted torus knots and generalized the criteria due to Clay and Watson and due to Ichihara and Temma.

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