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Gauss-Gassner Invariants

Knots in the Triangle (Knots in Washington XLII), North Carolina State University, April 29 - May 1, 2016

Abstract. In a "degree $d$ Gauss diagram formula" one produces a number by summing over all possibilities of paying very close attention to $d$ crossings in some $n$-crossing knot diagram while observing the rest of the diagram only very loosely, minding only its skeleton. The result is always poly-time computable as only $\binom{n}{d}$ states need to be considered. An under-explained paper by Goussarov, Polyak, and Viro [GPV] shows that every type $d$ knot invariant has a formula of this kind. Yet only finitely many integer invariants can be computed in this manner within any specific polynomial time bound.

I suggest to do the same as [GPV], except replacing "the skeleton" with "the Gassner invariant", which is still poly-time. One poly-time invariant that arises in this way is the Alexander polynomial (in itself it is infinitely many numerical invariants) and I believe (and have evidence to support my belief) that there are more.

Handout: GG.html, GG.pdf, GG.png.

Proposal: Summary.pdf, Detailed.pdf.

Sources: pensieve.

Talk video: