The Miyazawa polynomial for long virtual knots
type: Item
instance of: scholarly article
title: The Miyazawa polynomial for long virtual knots
zbMATH Open document ID: 1194.57018
author: Atsushi Ishii
author: Naoko Kamada
author: Seiichi Kamada
published in: Topology and its Applications
publication date: 2009-12-03T00:00:00Z
review text: In this article, a Miyazawa polynomial is defined for long virtual knots. A long virtual knot is essentially a single component one-one tangle that possibly contains virtual crossings. From any long virtual knot, we can recover a virtual knot by taking the closure, i.e. identifying the endpoints of the tangle. This defines a map from the equivalence classes of long virtual knots to the equivalence classes of virtual knots. The closure of a long virtual knot \(K\) is denoted as \( \hat{K} \). Unlike in the classical case, there is no inverse map. That is, selecting a point on a virtual knot and simply cutting produces a long virtual knot but the location of the point matters; different cut points can produce different long virtual knots. The Miyazawa polynomial \(M_K (A,t) \) of a knot \(K\) is an oriented version of the Jones polynomial and its states consist of bivalent graphs that are augmented by weight maps on the edges of the graphs. The contribution of the variable \(t\) is computed by evaluating the virtual writhe which is obtained by summing the Miyazawa index of each virtual crossing. The Miyazawa index assigns a weight to each crossing based on the values of edges in the state from the weight map. Here the authors define a related invariant, \( \Lambda_K (A,t) \), for a long virtual knot \(K\). This invariant has several properties which are detailed in the main result of the paper. The main results are as follows. Let \(K\) be a long virtual knot and let \( \hat{K}\) denote its closure. If \(K\) is a product (concatenation) of the long virtual knots \(K_1\) and \(K_2\) then \( \Lambda_K (A,t) = \Lambda_{K_1} (A,t) \Lambda_{K_2} (A,t) \). If \(M_{ \hat{K} } (A,t) \) is the Miyazawa polynomial of \( \hat{K} \) then \(M_{ \hat{K} } (A,t) = \frac{1}{2} ( \Lambda_K (A,t) + \Lambda_K (A, t^{-1}) )\). The Kauffman \(f\)-polynomial of \( \hat{K}\) can be recovered by evaluating \( \Lambda_K (A,1)\). Finally, the real crossing number and the virtual crossing numbe…
reviewed by: Heather A. Dye
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 5641772
zbMATH Keywords: long virtual knot
zbMATH Keywords: Miyazawa polynomial
zbMATH Keywords: Kauffman \(f\)-polynomial
MaRDI profile type: Publication
full work available at URL: j.topol.2009.04.058
OpenAlex ID: W2032233742
cites work: Minimal surface representations of virtual knots and links
cites work: Finite-type invariants of classical and virtual knots
cites work: A polynomial invariant for knots via von Neumann algebras
cites work: Hecke algebra representations of braid groups and link polynomials
cites work: Detecting Non-Triviality of Virtual Links
cites work: An index of an enhanced state of a virtual link diagram and Miyazawa polynomials
cites work: A 2-variable polynomial invariant for a virtual link derived from magnetic graphs
cites work: State models and the Jones polynomial
cites work: Virtual knot theory
cites work: Q4821340
cites work: LONG VIRTUAL KNOTS AND THEIR INVARIANTS
cites work: MAGNETIC GRAPHS AND AN INVARIANT FOR VIRTUAL LINKS
cites work: ALEXANDER GROUPS OF LONG VIRTUAL KNOTS
DOI: 10.1016/J.TOPOL.2009.04.058
description: scientific article; zbMATH DE number 5641772