2012/11/02 by Peter M. Johnson, Johnson, Peter M., Sóstenes Lins +1
Mathematics · #Geometric and Algebraic Topology #Advanced Combinatorial Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1211.0422
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.