SET-THEORETIC YANG–BAXTER SOLUTIONS VIA FOX CALCULUS

2006 ◽  
Vol 15 (08) ◽  
pp. 949-956 ◽  
Author(s):  
J. SCOTT CARTER ◽  
MASAHICO SAITO
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Baxter Equation ◽  
Group Representations ◽  
Group Automorphisms ◽  
Free Group Automorphisms

We construct solutions to the set–theoretic Yang–Baxter equation using braid group representations in free group automorphisms and their Fox differentials. The method resembles the extensions of groups and quandles.

scholarly journals Algebraic numbers, free group automorphisms and substitutions on the plane

2011 ◽  
Vol 363 (9) ◽  
pp. 4651-4699 ◽  
Author(s):  
Pierre Arnoux ◽  
Maki Furukado ◽  
Edmund Harriss ◽  
Shunji Ito
Keyword(s):  
Free Group ◽  
Algebraic Numbers ◽  
Group Automorphisms ◽  
Free Group Automorphisms

FERMIONS AND LINK INVARIANTS

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 493-532 ◽  
Author(s):  
L. Kauffman ◽  
H. Saleur
Keyword(s):  
Braid Group ◽  
Degrees Of Freedom ◽  
Knot Theory ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Group Representations ◽  
R Matrix ◽  
Link Invariants ◽  
Linear Representations ◽  
State Models

This paper deals with various aspects of knot theory when fermionic degrees of freedom are taken into account in the braid group representations and in the state models. We discuss how the Ř matrix for the Alexander polynomial arises from the Fox differential calculus, and how it is related to the quantum group Uqgl(1,1). We investigate new families of solutions of the Yang Baxter equation obtained from "linear" representations of the braid group and exterior algebra. We study state models associated with Uqsl(n,m), and in the case n=m=1 a state model for the multivariable Alexander polynomial. We consider invariants of links in solid handlebodies and show how the non trivial topology lifts the boson fermion degeneracy that is present in S3. We use "gauge like" changes of basis to obtain invariants in thickened surfaces Σ×[0,1].

Free Group Automorphisms and Knotted Tori in S4

1997 ◽  
Vol 06 (01) ◽  
pp. 95-103 ◽  
Author(s):  
Daniel S. Silver
Keyword(s):  
Free Group ◽  
Commutator Subgroup ◽  
Sufficient Conditions ◽  
Simple Construction ◽  
Finitely Generated ◽  
Seifert Manifolds ◽  
Group Automorphisms ◽  
Free Group Automorphisms ◽  
Finitely Presented

Let ϕ be an automorphism of a finitely generated free group F, w and element of F, and H the subgroup of F generated by the orbit {ϕn (w), |n ∈ Z}. We describe sufficient conditions ensuring that H is non-finitely generated. Using this we give a simple construction of tori T embedded in S4 in such a way that the commutator subgroup of π1(S4 - T) is finitely generated but not finitely presented. Such tori have no minimal Seifert manifolds. An example of an embedded torus with the latter property was given recently by T. Maeda using different methods.

scholarly journals North–South dynamics of hyperbolic free group automorphisms on the space of currents

2019 ◽  
Vol 11 (02) ◽  
pp. 427-466 ◽  
Author(s):  
Martin Lustig ◽  
Caglar Uyanik
Keyword(s):  
Free Group ◽  
Outer Automorphism ◽  
Train Track ◽  
Geodesic Currents ◽  
Group Automorphisms ◽  
Free Group Automorphisms

Let [Formula: see text] be a hyperbolic outer automorphism of a non-abelian free group [Formula: see text] such that [Formula: see text] and [Formula: see text] admit absolute train track representatives. We prove that [Formula: see text] acts on the space of projectivized geodesic currents on [Formula: see text] with generalized uniform North-South dynamics.

scholarly journals Mapping Tori of Free Group Automorphisms are Coherent

1999 ◽  
Vol 149 (3) ◽  
pp. 1061 ◽  
Author(s):  
Mark Feighn ◽  
Michael Handel
Keyword(s):  
Free Group ◽  
Mapping Tori ◽  
Group Automorphisms ◽  
Free Group Automorphisms

scholarly journals BRAID GROUP REPRESENTATIONS ARISING FROM THE YANG–BAXTER EQUATION

2010 ◽  
Vol 19 (04) ◽  
pp. 525-538 ◽  
Author(s):  
JENNIFER M. FRANKO
Keyword(s):  
Group Algebra ◽  
Braid Group ◽  
Roots Of Unity ◽  
Baxter Equation ◽  
Group Representations ◽  
Link Invariant

This paper aims to determine the images of the braid group under representations afforded by the Yang–Baxter equation when the solution is a non-trivial 4 × 4 matrix. Making the assumption that all the eigenvalues of the Yang–Baxter solution are roots of unity, leads to the conclusion that all the images are finite. Using results of Turaev, we have also identified cases in which one would get a link invariant. Finally, by observing the group algebra generated by the image of the braid group sometimes factor through known algebras, in certain instances we can identify the invariant as particular specializations of a known invariant.

scholarly journals Long turns, INP’s and indices for free group automorphisms

2015 ◽  
Vol 59 (4) ◽  
pp. 1087-1109 ◽  
Author(s):  
Thierry Coulbois ◽  
Martin Lustig
Keyword(s):  
Free Group ◽  
Group Automorphisms ◽  
Free Group Automorphisms
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