QUANTUM MECHANICS AND NON-ABELIAN THETA FUNCTIONS ...

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 29px yzm nFigure 12this computation using skein relations. Specifically, if three framed linksL,H,V in S 3 colored by V 2 coincide except in a ball where they look likein Figure 13, then their Reshetikhin-Turaev invariants, denoted by J L ,J H ,and J V satisfyJ L = tJ H + t −1 J Vif the two crossing strands come from different components, andJ L = ǫ(tJ H − t −1 J V )if the crossing strands come from the same component. Here ǫ is the signof the crossing.Additionally if a link component bounds a disk inside a balldisjoint from the rest of the link, then it is replaced by a factor of t 2 + t −2 .L H VFigure 13Using these skein relations we introduce a different type of skein module.For this, let M be an orientable 3-dimensional manifold. Consider the freeC[t,t −1 ]-module with basis the isotopy classes of framed oriented links inM, then factor it by the skein relationsL = tH + t −1 Vif the two crossing strands come from different components, andL = ǫ(tH − t −1 V )if the crossing strands come from the same component (with the same conventionfor ǫ as above) where the links L,H,V are the same except in anembedded ball in M and inside this ball they look like in Figure 13. Also,impose that any trivial link component that lies inside a ball disjoint fromthe rest of the link is replaced by a factor of t 2 + t −2 . We call the moduleobtained this way the Reshetikhin-Turaev skein module and denote it byRT t (M).Like before, if M = Σ g × [0,1] then the homeomorphismΣ g × [0,1] ∪ Σg Σ g × [0,1] = Σ g × [0,1]