Kauffman bracket skein module of two families of Seifert manifolds

Abstract

We compute the Kauffman bracket skein modules of Seifert manifolds ${\Sigma }_{0,1}((k_1,1),(k_2,1))$ and ${\Sigma }_{0,0}((k_1,1),(k_2,1),(k_3,1))$ by providing presentations of them. From the obtained presentations, we show that the Kauffman bracket skein modules of ${\Sigma }_{0,1}((k_1,1),(k_2,1))$ are free with infinitely many generators when $k_1,k_2\ge 1$ and that of ${\Sigma }_{0,0}((k_1,1),(k_2,1),(k_3,1))$ are finitely generated when $k_1,k_2,k_3\ge 2$. We also show that the empty link in either case is not trivial.

29 pages, 14 figures