We compute the HOMFLYPT skein module of the lens spaces L p,1 and present a free basis of this module for each p.

Diagrams and Reidemeister moves for links in orientable Seifert manifolds are introduced. Using these diagrams, we compute the Kauffman bracket skein modules of prism manifolds with first homology of order 4. In particular, we show that these modules are free.

Abstract. Diagrams and Reidemeister moves for links in a twisted S 1 -bundle over an unorientable surface are introduced. Using these diagrams, we compute the Kauffman Bracket Skein Module (KBSM) of RP 3 ♯RP 3 . In particular, we show that it has torsion. We also present a new computation of the KBSM of S 1 × S 2 and the lens spaces L(p, 1).