Topologically slice knots with nontrivial Alexander polynomial
Matthew Hedden, Charles Livingston, Daniel Ruberman
TL;DR
This paper demonstrates that the subgroup of topologically slice knots with nontrivial Alexander polynomial is infinitely generated, revealing complex structure in knot concordance and related bordism groups.
Contribution
It proves the infinite generation of the quotient of certain knot concordance subgroups and constructs links with topological but not smooth concordance.
Findings
C_T/C_D is infinitely generated
Existence of links topologically but not smoothly concordant to boundary links
Uncovering complex structure in rational spin bordism group
Abstract
Let C_T be the subgroup of the smooth knot concordance group generated by topologically slice knots and let C_D be the subgroup generated by knots with trivial Alexander polynomial. We prove the quotient C_T/C_D is infinitely generated, and uncover similar structure in the 3-dimensional rational spin bordism group. Our methods also lead to the construction of links that are topologically, but not smoothly, concordant to boundary links.
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