Homotopy ribbon concordance and Alexander polynomials

We show that if a knot J in the 3-sphere is homotopy ribbon concordant to a knot K then the Alexander polynomial of K divides the Alexander polynomial of J.

The notion of homotopy ribbon concordance is a natural homotopy group analogue of the notion of smooth ribbon concordance introduced by C. Gordon [Gor81]: J is (smoothly) ribbon concordant to K, written J ≥ sm K, if there is a smooth concordance from J to K such that the restriction of the projection map S 3 ×I → I to C yields a Morse function on C without minima. The exterior of such a concordance admits a handle decomposition relative to X J with only 2-and 3handles, from which it is easy to see that the induced map on fundamental groups π 1 (X J ) → π 1 (X C ) is surjective. Gordon also showed that π 1 (X K ) → π 1 (X C ) is injective. Thus a ribbon concordance is a homotopy ribbon concordance.
The Alexander polynomial ∆ J (t) ∈ Z[t ±1 ] of an oriented knot J is by definition the order of the first homology H 1 (X J ; Z[t ±1 ]) of the infinite cyclic cover of J. Equivalently this is the determinant of tV − V T , for any Seifert matrix V for J.
For ≥ sm instead of ≥ top , Theorem 1.1 is a consequence of a more general theorem of Gilmer [Gil84]. However Gilmer's proof does not extend to the topological locally flat category.
We want to explain a simple proof of Theorem 1.1. We expect that our argument can be generalised to twisted Alexander polynomials [KL99a,KL99b,HKL10] and higher order Alexander polynomials [Coc04], at least those defined over a UFD, provided one uses a unitary representation that extends over the ribbon concordance exterior. Our proof can also be generalised to concordances between knots in homology spheres. Having not found a convincing application, we have not carried out either of these generalisations in this short note.
A number of articles have recently appeared on the relation of ribbon concordance to Heegaard-Floer and Khovanov homology [Zem19, LZ19, MZ19, JMZ19, Sar19]. These techniques of course do not apply to locally flat concordances. For those working in the smooth category, we thought it might be of interest to show how to establish, with minimal machinery, the existence of two concordant knots that are not ribbon concordant.
Remark 1.2. It is straightforward to apply Theorem 1.1 to construct examples of concordant knots that are not homotopy ribbon concordant. For instance (this example was given by Gordon [Gor81], but with a different proof), let K be a trefoil and let J be the figure eight knot. Then K# − K and J# − J are both slice, so are concordant. But the Alexander polynomials are coprime, so there is no homotopy ribbon concordance between these knots.
Remark 1.3. The condition that π 1 (X K ) → π 1 (X C ) is injective is not needed anywhere in our proof of Theorem 1.1.
Gordon conjectured that ribbon concordance gives a partial order on knots. This conjecture is still open: in order to prove it, one would have to show that if J is ribbon concordant to K and K is ribbon concordant to J, then K and J are isotopic.
There is a concordance C with π 1 (X C ) ∼ = Z from the unknot U to K for every Alexander polynomial one knot K. So in order to make the analogous conjecture that ≥ top is a partial order, we have included that π 1 (X K ) → π 1 (X C ) is injective in the definition. Thus, the concordance C is not a homotopy ribbon concordance.
Conjecture 1.4. The relation ≥ top is a partial order on the set of knots.
Acknowledgements. We would like to thank Arunima Ray and the Max Planck Institute for Mathematics in Bonn.

Injection and surjection of Alexander modules
Theorem 1.1 will follow immediately from the next two propositions on Alexander modules.
Proposition 2.1. If C is a ribbon concordance from J to K, then the induced map is surjective.
Proof. Consider the following commutative diagram Since the middle map is an epimorphism we see that map on the left is an epimorphism. By the Hurewicz theorem, the induced map on homology is an epimorphism. But by the Shapiro lemma the homology groups H 1 (ker(π 1 (X J ) → Z); Z) and H 1 (ker(π 1 (X C ) → Z); Z) are precisely the twisted homology groups H 1 (X J ; Z[t ± ]) and H 1 (X C ; Z[t ± ]) respectively.
Here is another proof using homological algebra, for which generalisation to twisted coefficients would be easier.
Since π = π 1 (X C ), we have H 1 (X C ; Zπ) = 0 and H 0 (X C ; Zπ) ∼ = Z. Since π 1 (X J ) → π is surjective, the induced cover of X J is connected, so H 0 (X J ; Zπ) ∼ = Z and the map to H 0 (X C ; Zπ) is an isomorphism. We deduce that Next, apply the universal coefficient spectral sequence for homology Corollary 2.2. The orders of the homologies satisfy: In the next proposition we shall make use of the next lemma. The proof is a straightforward check and is omitted.
is well-defined and is an isomorphism of Z[t ±1 ]-cochain complexes.
Proposition 2.4. Suppose that J is homotopy ribbon concordant to K. Then the induced map Proof. We show that H 2 (X C , X K ; Z[t ±1 ]) = 0. First, the fact that X K → X C induces a Z-homology isomorphism implies that H i (X C , X K ; Z) = 0 for all i. By chain homotopy lifting [COT03, Proposition 2.10] this implies that H i (X C , X K ; Q(t)) = 0 for all i. Since commutative localisation is flat, this implies in particular that Now by Poincaré-Lefschetz duality, As above write π := π 1 (X C ). Now by Lemma 2.3. We can compute the right hand side of this using the universal coefficient spectral sequence for cohomology [Lev77, Theorem 2.3], which combined with the equation above gives a spectral sequence . We shall show that all the terms on the 2-line p + q = 2 vanish. First, since ]) = 0. This completes the proof that all the terms on the 2-line vanish, so we see that H 2 (X C , X K ; Z[t ±1 ]) ∼ = H 2 (X C , X J ; Z[t ±1 ]) = 0 which implies that H 2 (X C , X K ; Z[t ±1 ]) = 0 as desired. It then follows from the long exact sequence of the pair (X C , X K ) that the map is injective.
Corollary 2.5. The orders of the homologies satisfy: 3. Proof of Theorem 1.1 By Corollary 2.5, we have that ∆ K = ord H 1 (X K ; Z[t ±1 ]) divides ∆ C := ord H 1 (X C ; Z[t ±1 ]). That is, ∆ C = ∆ K · p for some p ∈ Z[t ±1 ]. On the other hand, by Corollary 2.2, for some q ∈ Z[t ±1 ] we have that ∆ C · q = ∆ J . Therefore ∆ J = ∆ C · q = ∆ K · p · q and so ∆ K | ∆ J as desired. This completes the proof of Theorem 1.1.