On the impossibility of a generalization of the HOMFLY-polynomial to labelled oriented graphs
type: Item
instance of: scholarly article
title: On the impossibility of a generalization of the HOMFLY-polynomial to labelled oriented graphs
zbMATH Open document ID: 0874.57001
author: Antje Christensen
author: Stephan Rosebrock
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1997-11-02T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73388
review text: Let \(D=\langle x_1,x_2,...,x_n|R_1,R_2,...,R_n\rangle\) be a group presentation, where each relator is of the form \(x_ix_j=x_jx_k\). We can assign a labelled oriented graph \(\Gamma(D)\) to \(D\) by defining a vertex \(i\) for every generator \(x_i\) and an oriented edge from \(i\) to \(k\) labelled by \(j\) for the relator \(x_ix_j=x_jx_k\). For a regular diagram of an oriented link \(L\) in \({\mathbb R}^3\), a Wirtinger presentation \(D\) of a group \(\pi_1({\mathbb R}^3\smallsetminus L)\) has the form of a labelled oriented graph \(\Gamma(D)\) with each component a circle or a single vertex. Let \(\mathcal C\) be the set of the labelled oriented graphs with each component a circle or a single vertex. We note that a local change of a regular diagram of a link \(L\) induces a local change of a Wirtinger presentation \(D\) of \(\pi_1({\mathbb R}^3\smallsetminus L)\), and also a local change of \(\Gamma(D)\). This implies Reidemeister moves and HOMFLY polynomial have algebraic analogues for \(\mathcal C\). The author shows that an algebraic generalization of HOMFLY polynomial to \(\mathcal C\) is almost trivial. More precisely if HOMFLY polynomial of a labelled oriented graph \(\Gamma\) in \(\mathcal C\) is an invariant under Reidemeister moves, then it only detects the parity of the number of components of \(\Gamma\).
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 1013900
zbMATH Keywords: HOMFLY polynomial
zbMATH Keywords: labelled oriented graph
zbMATH Keywords: link
zbMATH Keywords: Wirtinger presentation
reviewed by: Akira Yasuhara
MaRDI profile type: Publication
OpenAlex ID: W2049634702
cites work: Knots
cites work: Q4280844
cites work: Some remarks on a problem of J. H. C. Whitehead
cites work: On the Asphericity of Ribbon Disc Complements
cites work: Q4000283
cites work: A polynomial invariant of oriented links
cites work: Q4107635
cites work: Q3200251
cites work: ON THE REALIZATION OF WIRTINGER PRESENTATIONS AS KNOT GROUPS
DOI: 10.5802/AFST.833
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description: scientific article; zbMATH DE number 1013900