Narrow Results

The normalized Yamada polynomial, , is a polynomial invariant in variable A for θ-curves. In this work, we show that the coefficients of which is obtained by replacing A with e^x = ∑ xⁿ/n! are finite-type invariants for θ-curves although the coefficients of original are not finite-type. A similar result can be obtained in the case of Yokota polynomial for θ-curves.

We enumerate all the prime θ-curves with up to seven crossings to confirm a table made by Litherland. These θ-curves are mutually distinguished by the Yamada polynomial.

In this paper, we compute the graph skein algebra of the punctured disk with two holes. Then, we apply the graph skein techniques developed here to establish necessary conditions for a spatial graph to have a symmetry of order p, where p is a prime. The obstruction criteria introduced here extend some results obtained earlier for symmetric spatial graphs.

We present formulae for computing the Yamada polynomial of spatial graphs obtained by replacing edges of plane graphs, such as cycle-graphs, theta-graphs, and bouquet-graphs, by spatial parts. As a corollary, it is shown that zeros of Yamada polynomials of some series of spatial graphs are dense in a certain region in the complex plane, described by a system of inequalities. Also, the relation between Yamada polynomial of graphs and the chain polynomial of edge-labeled graphs is obtained.

In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These virtual graphs are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the $S$-polynomial, and formulate the $𝔰𝔩(N)$ Penrose polynomial for non-cubic graphs, giving contraction–deletion relations. The $S$-polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the $S$-polynomial at squares of integers.

Let be a spatial graph in the 3-sphere which covers a graph in the lens space. We introduce a combinatorial representation of which reflects the symmetry of the spatial graph. This representation involves a ribbon n-graph and the generator of the center of the n-braid group. We apply this result to study the Yamada polynomial of a class of lens spatial graphs.

As an extension of the Yamada polynomial for spatial graphs, we construct a polynomial invariant for virtual graphs. Since a virtual link can be regarded as a virtual graph having no vertices, the polynomial is an invariant for virtual links. We show that the invariant is useful for detecting non-classicality of a virtual knot.

For any graph G we define bigraded cohomology groups whose graded Euler characteristic is a multiple of the Yamada polynomial of G.