11 research outputs found
Symbolic 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 Z_p 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.Comment: 8 pages. Version 2 has typo corrections and changes suggested by
referee. To appear in J. Symbolic Compu
Generalized 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.Comment: 11 pages. Version 3 includes a correction to the square/granny knot
example. To appear in Can. Bull. Mat
A Survey of Quandle Ideas
This article surveys many aspects of the theory of quandles which
algebraically encode the Reidemeister moves. In addition to knot theory,
quandles have found applications in other areas which are only mentioned in
passing here. The main purpose is to give a short introduction to the subject
and a guide to the applications that have been found thus far for quandle
cocycle invariants.Comment: Submitted to conference proceedings; embarrassing misspellings of
various names corrected. Many apologies and thanks to readers who pointed out
correction
A Survey of Racks and Quandles: Some recent developments
This short survey contains some recent developments of the algebraic theory
of racks and quandles. We report on some elements of representation theory of
quandles and ring theoretic approach to quandles.Comment: 13 pages. ICART 2018. To appear in Algebra Colloquiu
Set-theoretic solutions of the Yang-Baxter equation, RC-calculus, and Garside germs
Building on a result by W. Rump, we show how to exploit the right-cyclic law
(x.y).(x.z) = (y.x).(y.z) in order to investigate the structure groups and
monoids attached with (involutive nondegenerate) set-theoretic solutions of the
Yang-Baxter equation. We develop a sort of right-cyclic calculus, and use it to
obtain short proofs for the existence both of the Garside structure and of the
I-structure of such groups. We describe finite quotients that exactly play for
the considered groups the role that Coxeter groups play for Artin-Tits groups