Cyclotomic invariants for links
type: Item
instance of: scholarly article
title: Cyclotomic invariants for links
zbMATH Open document ID: 0668.57013
author: Tsuyoshi Kobayashi
author: Hitoshi Murakami
author: Jun Murakami
published in: Proceedings of the Japan Academy. Series A
publication date: 1988-01-01T00:00:00Z
review text: In this paper the authors introduce numerical invariants \(\tilde T_ N(L)\) of a link L, called cyclotomic invariants. They are defined by using the partition function of an exactly solvable N-state IRF-model on a two-dimensional square lattice \((N=1,2,...)\) associated with the graph of a link diagram of a link L. Therefore, \(\tilde T_ N(L)\) is closely related to the Jones polynomial \(V_ L(t)\) of L. In fact, the authors show that \(\tilde T_ N(L)\) can be described by using the Goeritz matrix and the Seifert matrix of L. The main theorem is Theorem 3. (1) \(\tilde T_ 2(L)=(V_ L(\sqrt{i}))^ 2\), (2) \(\tilde T_ 3(L)=V_ L(e^{\pi i/3})\), (3) \(| \tilde T_ p(L)|^ 2=(\sqrt{p})^{\beta_ p(L)}\), where \(\beta_ p(L)\) is the first Betti number of the double covering of L with coefficients in \({\mathbb{Z}}/p{\mathbb{Z}}\), p being an odd prime. Furthermore, it is shown that \(\tilde T_ 2(L)\) and \(\tilde T_ 3(L)\) are also determined by the Kauffman polynomial \(F_ L(a,x)\) and the Q-polynomial \(Q_ L(x)\) (Corollary).
reviewed by: K. Murasugi
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 4094401
zbMATH Keywords: Betti number of double covering
zbMATH Keywords: cyclotomic invariants
zbMATH Keywords: N-state IRF- model
zbMATH Keywords: link diagram
zbMATH Keywords: Jones polynomial
zbMATH Keywords: Goeritz matrix
zbMATH Keywords: Seifert matrix
zbMATH Keywords: Kauffman polynomial
zbMATH Keywords: Q-polynomial
MaRDI profile type: Publication
full work available at URL: pjaa.64.235
OpenAlex ID: W1997678973
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DOI: 10.3792/PJAA.64.235
description: scientific article; zbMATH DE number 4094401