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

We generalize Ngs 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 Ngs three-variable knot contact homology. Our main tool is Lins 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.