Narrow Results

A. S. Lipson constructed two state models yielding the same classical link invariant obtained from the Kauffman polynomial F(a, u). In this paper, we apply Lipson's state models to marked graph diagrams of surface-links, and observe when they induce surface-link invariants.

In this paper, we give several simple criteria to detect possible periods and linking numbers for a given virtual link. We investigate the behavior of the generalized Alexander polynomial $Z_L$ of a periodic virtual link $L$ via its Yang–Baxter state model given in [L. H. Kauffman and D. E. Radford, Bi-oriented quantum algebras and a generalized Alexander polynomial for virtual links, in Diagrammatic Morphisms and Applications, Contemp. Math.318 (2003) 113–140, arXiv:math/0112280v2 [math.GT] 31 Dec 2001].

In this study, we introduce new criteria for a given link to detect the non-$p$-periodicity for a prime $p\ge 5$. For this we use a congruence of the quantum ${𝔰𝔩}_N$-invariant of periodic links. Our result is naturally consistent with Traczyk’s criterion and Przytycki’s criterion, and can be regarded as a generalization of these classical criteria. We shall give computational results of all the (alternating and non-alternating) knots of crossing number $\le 16$ for their non-$p$-periodicity ($p\ge 5$). The computational results also show that our criteria has somewhat different nature with the classical criterion of Murasugi.

A state model for Kauffman polynomial of Dubrovnik-version is described using Gauss diagrams. Based on the state model, the Gauss diagram formulae for Vassiliev invariants are given from the coefficients of Kauffman polynomial following the method of Chmutov and Polyak. Some arrow diagram identities are given to simplify the Gauss diagram formulae of order 3, which give Polyak–Viro and Chmutov–Polyak formulae for the Vassiliev invariant of order 3. The models of Kauffman polynomial and HOMFLY-PT polynomial give different Gauss diagram expressions when specializing in Jones polynomial.

R. Bott defined two combinatorial invariants for finite cell complexes in the 1950's [2]. As described in [3], they fit beautifully into a state model framework. Recently, it has been observed that for finite graphs, the coefficients of this polynomial are the same set of invariants as defined by H. Whitney in [5]. In section 1, we define the Bott-Whitney polynomial and prove some of its basic properties. In section 2, we show that for a planar connected graph, the Bott-Whitney polynomial is essentially the chromatic polynomial of its dual graph. In section 3, an interpretation of the coefficients of the Bott-Whitney polynomial is given following Whitney [5]. In section 4, we study more general Bott polynomials.