Spin models, association schemes and the Nakanishi--Montesinos conjecture
type: Item
instance of: scholarly article
title: Spin models, association schemes and the Nakanishi--Montesinos conjecture
zbMATH Open document ID: 1019.57004
author: Pierre Manchess
author: Stéphan Ceroi
published in: European Journal of Combinatorics
publication date: 2003-03-18T00:00:00Z
review text: The Nakanishi-Montesinos (NM) conjecture asserts that any link can be reduced to an unknotted collection of unknots by performing 3-transformations. The authors introduce the notion of the NM-spin model, which gives a link invariant preserved by 3-transformations and prove that the Boser-Mesner algebra generated by the spin model is less or equals 4. It is proved that for dimensions 1 and 2 there is no such spin model but for dimension 3 there exists a unique one for which the link invariant corresponds to a specialization of the Kauffman polynomial but does not distinguish trivial links from non-trivials ones (hence it cannot disprove the NM conjecture). A family of NM-spin models are given for dimension 4. A number of proofs are omitted but they can be found in \textit{S. Ceroi} and \textit{P. Manches} [Spin models, association schemes and the Nakanishi-Montesinos conjecture, Technical Report 99-149, LIRMM 161 rue Ada 34392 Montpellier, France October 1999].
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 1884169
zbMATH Keywords: Nakanishi-Montesinos conjecture
zbMATH Keywords: 3-transformations
zbMATH Keywords: NM-spin model
zbMATH Keywords: link invariant
zbMATH Keywords: Boser-Mesner algebra
reviewed by: Jorge Luis Ramírez Alfonsín
MaRDI profile type: Publication
full work available at URL: eujc.2000.0461
OpenAlex ID: W2036225780
Wikidata QID: Q123179052
cites work: Q3218140
cites work: Q4154891
cites work: Q3992965
cites work: Knots
cites work: Graph invariants related to statistical mechanical models: Examples and problems
cites work: Strongly regular graphs
cites work: Strongly regular graphs and spin models for the Kauffman polynomial
cites work: On spin models, triply regular association schemes, and duality
cites work: Q4841307
cites work: Q4892002
cites work: On knot invariants related to some statistical mechanical models
cites work: GENERALIZED SPIN MODELS
cites work: Some link-polynomial relations
cites work: A polynomial invariant for unoriented knots and links
cites work: Q3341550
DOI: 10.1006/EUJC.2000.0461
description: scientific article; zbMATH DE number 1884169