A state sum invariant of tangles in surfaces

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 $\mathbb{Z}[u]$, 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 $\mathbb{R}^3$. 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.