Quantum equations for knots

Abstract

This paper contains linear systems of equations which can distinguish knots without knot invariants. Let $M_n$ be the topological moduli space of all n-component string links and such that a fixed projection into the plane is an immersion. If a string link is the product of some string link diagram $T$ and the parallel n-cable of a framed long knot diagram $D$, then there is a canonical arc $push$ in $M_n$, defined by pushing $T$ through the n-cable of $D$. In this paper we apply the combinatorial 1-cocycles from the HOMFLYPT and Kauffman polynomials in $M_n$ with values in the corresponding skein modules to this canonical arc in $M_n$. Some of the 1-cocycles lead to linear systems of equations in the skein modules, for each couple of diagrams $D$ and $D'$. If the system has no solution in the Laurent polynomials then $D$ and $D'$ represent different knots. We give first examples where we distinguish knots without any knot invariants. In particular, we distinguish the knot $9_{42}$ from its mirror image with equations coming from the HOMFLYPT polynomial. Notice that the knot $9_{42}$ and its mirror image share the same HOMFLYPT polynomial. On the other hand, each solution of the system gives rather fine information about any regular isotopy which connects $D$ with $D'$.