The column group and its link invariants
type: Item
instance of: scholarly article
zbMATH Open document ID: 1244.57025
DOI: 10.1142/S0218216512500630
author: Johanna Hennig
author: Sam Nelson
publication date: 2012-04-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/0902.0028
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 6029061
zbMATH Keywords: biracks
zbMATH Keywords: link invariants
zbMATH Keywords: enhancements of counting invariants
MaRDI profile type: Publication
arXiv ID: 0902.0028
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OpenAlex ID: W2964189458
title: The column group and its link invariants
published in: Journal of Knot Theory and its Ramifications
review text: A \textit{birack} is a pair, denoted by \((X, B)\), of a set \(X\) and a map \(B : X\times X\to X\times X\) satisfying certain axioms which correspond to the Reidemeister moves. Special cases of biracks include quandles, racks, and so on. The birack counting invariants can be defined for a link in a similar way as in the case of quandles. The main results of the paper are to define the column group \(\mathrm{CG}(X)\) of a finite birack \((X, B)\), and the \textit{column group enhanced birack multiset invariant} \(\phi_{(X, B)}^{\mathrm{CG}, M}(L)\) and the column group enhanced birack polynomial invariant \(\phi_{(X, B)}^{\mathrm{CG}}(L)\) for an oriented link \(L\), which are enhancements of the birack counting invariants. In Example 4.5 and 4.6, the authors show effectiveness of their invariants.NEWLINENEWLINEThe map \(B=(B_1(x, y), B_2(x, y))\) is strongly invertible (cf.\ Definition 2.1) if it is (i) invertible, (ii) sideways invertible, and (iii) diagonally invertible. The condition (i) implies \(B\) is bijective, and it corresponds to the type II move. The conditions (ii), (iii) and the kink map in Definition 2.2 correspond to the type I move. The set-theoretic Yang-Baxter equation in Definition 2.3 corresponds to the type III move. Since the kink map \(\pi : X\to X\) of \((X, B)\) is not always the identity, a birack coloring for a link diagram is an invariant of a blackboard-framed link which is a pair of a \(c\)-component link \(L\) and a writhe vector \(\mathbf{w}\in \mathbb{Z}^c\). If there exists an integer \(N\geq 1\) such that \(\pi^N\) is the identity, then a birack coloring is stable for a writhe vector modulo \(N\). If \(N=1\), then a birack is a strong biquandle.NEWLINENEWLINEFor a finite birack \((X, B)\) with \(X=\{ x_1, \ldots, x_n\}\), the birack matrix of \((X, B)\) is defined as an \(n\times (2n)\) matrix \(M=[M_{B_1}|\;M_{B_2}]\) consisting of two \(n\times n\) blocks \(M_{B_1}\) and \(M_{B_2}\) where the \((i, j)\) entries of \(M_{B_1}\) an…
reviewed by: Teruhisa Kadokami
description: scientific article; zbMATH DE number 6029061