Cyclotomic invariants for links
In this paper the authors introduce numerical invariants \(\tilde T_ N(L)\) of a link L, called cyclotomic invariants. They are defined by using the partition function of an exactly solvable N-state IRF-model on a two-dimensional square lattice \((N=1,2,...)\) associated with the graph of a link diagram of a link L. Therefore, \(\tilde T_ N(L)\) is closely related to the Jones polynomial \(V_ L(t)\) of L. In fact, the authors show that \(\tilde T_ N(L)\) can be described by using the Goeritz matrix and the Seifert matrix of L. The main theorem is Theorem 3. (1) \(\tilde T_ 2(L)=(V_ L(\sqrt{i}))^ 2\), (2) \(\tilde T_ 3(L)=V_ L(e^{\pi i/3})\), (3) \(| \tilde T_ p(L)|^ 2=(\sqrt{p})^{\beta_ p(L)}\), where \(\beta_ p(L)\) is the first Betti number of the double covering of L with coefficients in \({\mathbb{Z}}/p{\mathbb{Z}}\), p being an odd prime. Furthermore, it is shown that \(\tilde T_ 2(L)\) and \(\tilde T_ 3(L)\) are also determined by the Kauffman polynomial \(F_ L(a,x)\) and the Q-polynomial \(Q_ L(x)\) (Corollary).
scientific article; zbMATH DE number 4094401