  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.
