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Homotopy ribbon concordance, Blanchfield pairings, and twisted Alexander polynomials. (English) Zbl 1520.57002

A locally flat annulus \(C\) in \(S^3 \times [0,1]\) with \(\partial C= - J \times \{0\} \cup K \times \{1\}\) is called a homotopy ribbon concordance from \(J\) to \(K\) if the inclusion induced maps from the fundamental groups of the exteriors of \(J\) and \(K\) to the fundamental group of the exterior of \(C\) are respectively surjective and injective. Such a concordance is the topological analogue of a ribbon concordance between knots, or smooth concordance without local minima, first discussed by C. McA. Gordon [Math. Ann. 257, 157–170 (1981; Zbl 0451.57001)]. I. Agol recently proved in [Comm. Am. Math. Soc. 2, 374–379 (2022; doi:10.1090/cams/15)] that ribbon concordance is a partial order on the set of knots in \(S^3\), but both anti-symmetry and transitivity remain open for homotopy ribbon concordance.
In this paper, the authors show that if there is a homotopy ribbon concordance from \(J\) to \(K\), then roughly speaking the Blanchfield pairing on \(J\) contains that of \(K\). More precisely, they show that there is a submodule \(G\) of \(H_1(X_J; \mathbb{Z}[t^{\pm1}])\), the Alexander module of \(J\), such that the Blanchfield pairing of \(J\) vanishes on \(G \times G\) and that the induced pairing on \(H_1(X_J; \mathbb{Z}[t^{\pm1}])/ G\) is isometric to the Blanchfield pairing of \(K\). This result generalizes previous work [S. Friedl and M. Powell, Arch. Math. 115, No. 6, 717–725 (2020; Zbl 1459.57008)] giving an obstruction to homotopy ribbon concordance coming from Alexander polynomials. From this result, the authors derive more computable obstructions to homotopy ribbon concordance coming from the homology of the double branched cover of knots and the Levine-Tristram signatures, as well as proving that there exist infinitely many concordant knots with the same Seifert form that are pairwise not homotopy ribbon concordant. The authors also generalize the work of [loc. cit.] in a different direction by proving that a homotopy ribbon concordance from \(J\) to \(K\) implies that certain twisted Alexander polynomials of \(K\) divide the corresponding twisted Alexander polynomials of \(J\).

MSC:

57K10 Knot theory
57K14 Knot polynomials
57N70 Cobordism and concordance in topological manifolds

Software:

KnotInfo
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References:

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