Abstract
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.