Three-page approach to knot theory. Universal semigroup
type: Item
instance of: scholarly article
title: Three-page approach to knot theory. Universal semigroup
zbMATH Open document ID: 0958.57006
DOI: 10.1007/BF02467064
publication date: 2000-11-09T00:00:00Z
review text: In a previous paper [Funct. Anal. Appl. 33, No. 4, 260-269 (1999); translation from Funkts. Anal. Prilozh. 33, No. 4, 25-37 (1999; Zbl 0947.57005)], the author discussed three-page link diagrams as links embedded in a three-page book and a new way to encode nonoriented links in the three-page book by words in a \(12\)-letter alphabet and presented a finite set of local moves realizing the equivalence of link types. In the paper under review, the author generalizes the notion of three-page link diagram. More precisely, the author introduces the three-page tangle diagrams which form a finitely presented semigroup whose center classifies the nonoriented links in the three-space. The author also gives a finite presentation for the group of invertible elements of the semigroup and shows that the commutator subgroup of this group is isomorphic to the braid group of infinitely many strands. Finally, the Kauffman theorem on the existence and uniqueness of the \(L\)-polynomial invariant is stated in the three-page language.
reviewed by: Sang Youl Lee
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 1497238
zbMATH Keywords: three-page link
zbMATH Keywords: three-page tangle
zbMATH Keywords: three-page braid
zbMATH Keywords: band-link
zbMATH Keywords: semigroup
author: I. A. Dynnikov
MaRDI profile type: Publication
cites work: Embedding knots and links in an open book II. Bounds on arc index
cites work: Arc index and the Kauffman polynomial
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1497238