Some Thoughts On Polyak's Conjecture

Recently, M. Polyak introduced an operator D on formal sums of knot diagrams having the property that if F is a formal sum of Gauss diagrams, then D(F)=0 (modulo some relations). Polyak's Conjecture asserts that the converse of this result is also true. In this talk, we will carefully state Polyak's Conjecture and look at some interesting examples found by computer search.