Framed Cord Algebra Invariant of Knots in $S^1 \times S^2$

TL;DR

This paper extends Ng's framed knot contact homology to knots in $S^1 imes S^2$, establishing an equivalence with a generalized cord algebra and introducing a new variable potentially linked to Ng's three-variable invariant.

Contribution

It generalizes Ng's algebraic invariant from $S^3$ to $S^1 imes S^2$ and proves their equivalence, incorporating an additional variable in the cord algebra.

Findings

The framed cord algebra in $S^1 imes S^2$ matches the generalized Ng invariant.

Introduction of an extra variable in the cord algebra potentially relates to Ng's three-variable invariant.

Conjecture that the framed cord algebra is finitely generated for non-local knots.

Abstract

We generalize Ng's two-variable algebraic/combinatorial $0$-th framed knot contact homology for framed oriented knots in $S^3$ to knots in $S^1 \times S^2$, and prove that the resulting knot invariant is the same as the framed cord algebra of knots. Actually, our cord algebra has an extra variable, which potentially corresponds to the third variable in Ng's three-variable knot contact homology. Our main tool is Lin's generalization of the Markov theorem for braids in $S^3$ to braids in $S^1 \times S^2$. We conjecture that our framed cord algebras are always finitely generated for non-local knots.