Gauss-Gassner Invariants

Dror Bar-Natan

Knots in the Triangle (Knots kNot in Washington)
April 29 - May 1, 2016
North Carolina State University
Raleigh NC, USA

Organizers
Valentina Harizanov (GWU), Jozef H. Przytycki (GWU), Yongwu Rong (GWU), Radmila Sazdanovic (NCSU), Alexander Shumakovitch (GWU), Hao Wu (GWU)

Conference Homepage

Gauss-Gassner Invariants
by
Dror Bar-Natan
University of Toronto

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.

More at http://www.math.toronto.edu/~drorbn/Talks/NCSU-1604/.

Date received: April 17, 2016