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

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 Added by Shawn X. Cui
 Publication date 2014
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and research's language is English




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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.



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We establish a number of results about smooth and topological concordance of knots in $S^1times S^2$. The winding number of a knot in $S^1times S^2$ is defined to be its class in $H_1(S^1times S^2;mathbb{Z})cong mathbb{Z}$. We show that there is a unique smooth concordance class of knots with winding number one. This improves the corresponding result of Friedl-Nagel-Orson-Powell in the topological category. We say a knot in $S^1times S^2$ is slice (resp. topologically slice) if it bounds a smooth (resp. locally flat) disk in $D^2times S^2$. We show that there are infinitely many topological concordance classes of non-slice knots, and moreover, for any winding number other than $pm 1$, there are infinitely many topological concordance classes even within the collection of slice knots. Additionally we demonstrate the distinction between the smooth and topological categories by constructing infinite families of slice knots that are topologically but not smoothly concordant, as well as non-slice knots that are topologically slice and topologically concordant, but not smoothly concordant.
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