A state sum invariant of tangles in surfaces

الأصل · EN

In this paper we define a new state sum based on the regions defined by tangles on a surface which is an oriented closed surface with a finite number of open holes drilled. From this state sum we obtain an invariant of regular isotopy for the tangles named u-invariant. The values of the u-invariant are in Z[u], where u is a primitive fifth root of the unity. Various basic properties of the invariant are proved and discussed. It can be specialized to invariants of framed links in R³. Its categoric aspect is emphasized: composition of tangles in surfaces correspond to matrix multiplication. The values of the u-invariant conjugate under taking the mirror of the tangle.