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Register a new userSet Theoretic Yang-Baxter Solutions via Fox Calculus
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J. Scott Carter
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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.
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A homology theory is developed for set-theoretic Yang-Baxter equations, and knot invariants are constructed by generalized colorings by biquandles and Yang-Baxter cocycles.
We determine the Betti numbers for the (degenerate and normalized) set-theoretic Yang-Baxter (co)homology groups of cyclic biquandles and estimate their torsion subgroups. This partially settles the conjecture presented by Przytycki, Vojtechovsky, and Yang. We also obtain cocycles which are representatives of the elements of a basis for the free part of the cohomology group of a cyclic biquandle.
To every involutive non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation on a finite set $X$ there is a naturally associated finite solvable permutation group ${mathcal G}(X,r)$ acting on $X$. We prove that every primitive permutation group of this type is of prime order $p$. Moreover, $(X,r)$ is then a so called permutation solution determined by a cycle of length $p$. This solves a problem recently asked by A. Ballester-Bolinches. The result opens a new perspective on a possible approach to the classification problem of all involutive non-degenerate set-theoretic solutions.
In this paper we discuss and characterize several set-theoretic solutions of the Yang-Baxter equation obtained using skew lattices, an algebraic structure that has not yet been related to the Yang-Baxter equation. Such solutions are degenerate in general, and thus different from solutions obtained from braces and other algebraic structures. Our main result concerns a description of a set-theoretic solution of the Yang-Baxter equation, obtained from an arbitrary skew lattice. We also provide a construction of a cancellative and distributive skew lattice on a given family of pairwise disjoint sets.
Given a set-theoretic solution $(X,r)$ of the Yang--Baxter equation, we denote by $M=M(X,r)$ the structure monoid and by $A=A(X,r)$, respectively $A=A(X,r)$, the left, respectively right, derived structure monoid of $(X,r)$. It is shown that there exist a left action of $M$ on $A$ and a right action of $M$ on $A$ and 1-cocycles $pi$ and $pi$ of $M$ with coefficients in $A$ and in $A$ with respect to these actions respectively. We investigate when the 1-cocycles are injective, surjective or bijective. In case $X$ is finite, it turns out that $pi$ is bijective if and only if $(X,r)$ is left non-degenerate, and $pi$ is bijective if and only if $(X,r)$ is right non-degenerate. In case $(X,r) $ is left non-degenerate, in particular $pi$ is bijective, we define a semi-truss structure on $M(X,r)$ and then we show that this naturally induces a set-theoretic solution $(bar M, bar r)$ on the least cancellative image $bar M= M(X,r)/eta$ of $M(X,r)$. In case $X$ is naturally embedded in $M(X,r)/eta$, for example when $(X,r)$ is irretractable, then $bar r$ is an extension of $r$. It also is shown that non-degenerate irretractable solutions necessarily are bijective.
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