2010/02/02 by Micah Chrisman, Vassily Olegovich Manturov, Chrisman, Micah +1
Mathematics · #57M25 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1002.0539
openalex publication_date 2010/02/02 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
The present paper produces examples of Gauss diagram formulae for virtual\nknot invariants which have no analogue in the classical knot case. These\ncombinatorial formulae contain additional information about how a subdiagram is\nembedded in a virtual knot diagram. The additional information comes from the\nsecond author's recently discovered notion of parity. For a parity of flat\nvirtual knots, the new combinatorial formulae are Kauffman finite-type\ninvariants. However, many of the combinatorial formulae possess exotic\nproperties. It is shown that there exists an integer valued virtualization\ninvariant combinatorial formula of order n for every n (i.e. it is stable under\nthe map which changes the direction of one arrow but preserves the sign).\nHence, it is not of Goussarov-Polyak-Viro finite-type. Moreover, every\nhomogeneous Polyak-Viro combinatorial formula admits a decomposition into an\n"even" part and an "odd" part. For the Gaussian parity, neither part of the\nformula is of GPV finite-type when it is nonconstant on the set of classical\nknots. In addition, eleven new non-trivial combinatorial formulae of order 2\nare presented which are not of GPV finite-type.\n