Spin Models, Association Schemes and the Nakanishi–Montesinos Conjecture

✍ Scribed by Pierre Manches; Stéphan Ceroi

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 262 KB
• Volume: 23
• Category: Article
• ISSN: 0195-6698
• DOI: 10.1006/eujc.2000.0461

✦ Synopsis

A 3-transformation of a link is a local change which replaces two strings that are three times half twisted around each other by two untwisted strings (and vice versa). The Nakanishi-Montesinos (NM) conjecture asserts that this 3-transformation can unknot any link. We introduce the notion of the NM-spin model, which gives a link invariant preserved by 3-transformation. We try to classify such spin models and determine the corresponding link invariant. It is proved that the dimension of the Bose-Mesner algebra generated by the spin model is ≤4. For dimension 1 and 2, there is no such spin model, but for dimension 3, there exists a unique one. Its link invariant is a non-trivial specialization of the Kauffman polynomial, but does not distinguish trivial links from the others, and hence cannot disprove the NM conjecture. For dimension 4, we give a family of NM-spin models. The corresponding link invariant is identified and does not distinguish trivial links from the others. Strong regularity and triple regularity of the Bose-Mesner algebra and its fusions are studied.