Spin models, association schemes and the Nakanishi--Montesinos conjecture
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].
scientific article; zbMATH DE number 1884169