Classification of finite Alexander biquandles
نویسندگان
چکیده
We show that two finite Alexander biquandles M and M ′ are isomorphic iff there is an isomorphism of Z[s, t]-modules h : (1 − st)M → (1 − st)M ′ and a bijection g : Os(A) → Os(A ) between the s-orbits of sets of coset representatives of M/(1− st)M and M /(1− st)M ′ respectively satisfying certain compatibility conditions.
منابع مشابه
Virtual Knot Invariants from Group Biquandles and Their Cocycles
A group-theoretical method, via Wada’s representations, is presented to distinguish Kishino’s virtual knot from the unknot. Biquandles are constructed for any group using Wada’s braid group representations. Cocycle invariants for these biquandles are studied. These invariants are applied to show the non-existence of Alexander numberings and checkerboard colorings.
متن کاملAn isomorphism theorem for Alexander biquandles
We show that two Alexander biquandles M and M ′ are isomorphic iff there is an isomorphism of Z[s, t]-modules h : (1− st)M → (1− st)M ′ and a bijection g : Os(A) → Os(A ) between the s-orbits of sets of coset representatives of M/(1 − st)M and M /(1 − st)M ′ respectively satisfying certain compatibility conditions.
متن کاملTheory of Generalised Biquandles and Its Applications to Generalised Knots
In this thesis we present a range of different knot theories and then generalise them. Working with this, we focus on biquandles with linear and quadratic biquandle functions (in the quadratic case we restrict ourselves to functions with commutative coefficients). In particular, we show that if a biquandle is commutative, the biquandle function must have non-commutative coefficients, which ties...
متن کاملSymbolic computation with finite biquandles
A method of computing a basis for the second Yang-Baxter cohomology of a finite biquandle with coefficients in Q and Zp from a matrix presentation of the finite biquandle is described. We also describe a method for computing the Yang-Baxter cocycle invariants of an oriented knot or link represented as a signed Gauss code. We provide a URL for our Maple implementations of these algorithms.
متن کاملGeneralized quandle polynomials
We define a family of generalizations of the two-variable quandle polynomial. These polynomial invariants generalize in a natural way to eight-variable polynomial invariants of finite biquandles. We use these polynomials to define a family of link invariants which further generalize the quandle counting invariant.
متن کامل