Skein algebras of the solid torus and symmetric spatial graphs

Abstract

Abstract. We use the topological invariant of spatial graphs introduced by S. Yamada to find necessary conditions for a spatial graph to be periodic with a prime period. The proof of the main result is based on computing the Yamada skein algebra of the solid torus and then proving that it injects into the Kauffman bracket skein algebra of the solid torus.

Ribbon Graphs and Temperley-Lieb Algebra

“…The purpose of this section is to discuss the relationship between the two algebras Y n and τ n . In the case of skein modules of three-manifolds, we defined a homomorphism from the graph skein module to the Kauffman bracket skein module [1,2]. An analogous s of this homomorphism is defined here.…”

Graph Skein Modules and Symmetries of Spatial Graphs

“…It was proved in [6] that ϕ defines a map Φ from the graph skein module Y(F × I) to the Kauffman bracket skein module K(F × I), where F is any oriented surface. Obviously, Φ is a homomorphism of algebras.…”

“…By the reduction arguments used in the case of the annulus [6], we can prove that any graph diagram in F 0,3 can be written as a linear combination of finite disjoint unions of bouquets each of which has no contractible cycles. Arguments similar to the ones used to prove Lemma 3.3.2…”

“…in [6] would enable us to prove that the graph skein module of F 0,3 × I is generated by elements of type x i y j z k t ǫ , where i, j and k are nonnegative integers, and ǫ ∈ {0, 1}.…”

“…Marui [12], used the Yamada polynomial to study the periodicity of spatial graphs with wrapping number 1 or 2. In [6], we used the criteria of link periodicity introduced by Murasugi [13], Przytyki [15] and Traczyk [17] to obtain a generalization of Marui's result. We proved the following [6]:…”