Bordered HF and Fox-Milnor for slice fibered knots

Problem: Categorify the Fox-Milnor criterion for slice fibered knots.

If $K\subset S^3$ is a slice knot (either topologically or smoothly) then its Alexander polynomial satisfies $$\Delta_K(t) = f(t)f(t^{-1})$$ for some $f\in \mathbb{Z}[t,t^{-1}]$. This is called the Fox-Milnor criterion.

Knot Floer homology is said to categorify the Alexander polynomial; more precisely, its graded Euler characteristic satisfies $$\chi(\widehat{\mathit{HFK}}(K)) = \bigoplus_{m,a}(-1)^m t^a \dim \widehat{\mathit{HFK}}(K)  = \Delta_K(t).$$ It is therefore natural to ask (the admittedly vague question of) whether there are extra features of the knot Floer homology of slice knots which recover the Fox-Milnor criterion for the Alexander polynomial. One would hope moreover that these features provide a stronger obstruction to slicing a knot than the classical Fox-Milnor criterion. This is one thing that could be meant by the charge to categorify Fox-Milnor. 

It's not at all clear however what these extra features might be. The most naive guess is that for $K$ slice we have $$\widehat{\mathit{HFK}}(K) \cong V \otimes V^*$$ for some bi-graded vector space $V$. One way to build slice knots is by taking the connected sum of any knot with the orientation-reverse of its mirror. In this case, our naive guess is true by the Kunneth formula in knot Floer homology, but it does not hold in general. In fact, it is not even true that $\dim \widehat{\mathit{HFK}}(K)$ needs to be a perfect square when $K$ is slice. Noting that perfect squares leave a remainder of 1 mod 8, one might hope that at least $$\dim \widehat{\mathit{HFK}}(K)\equiv 1 \,\mathrm{mod}\,8$$ when $K$ is slice, but this is not true either as shown by Dunfield-Gong-Hockenhull-Marengon-Willis (not even for symmetric unions). The authors do conjecture that the above dimension is always 1 mod 4 for slice knots, and back it up with experimental evidence. But even if that's true, it does not recover Fox-Milnor for the Alexander polynomial.

Below is another potential route to categorifying Fox-Milnor via bordered Heegaard Floer invariants. 

Suppose $F^\circ$ is a Seifert surface for $K\subset S^3$ and let $F$ be the capped-off surface in $S^3_0(K)$. The complement $$W = S^3_0(K)\setminus \nu(F)$$ of a regular neighborhood of $F$  is naturally a bordered 3-manifold with boundary $\partial W = -F \sqcup F$. The work of Lipshitz-Ozsvath-Thurston assigns to $W$ a certain bimodule $\mathcal{F}_{DA}(W)$ over a dg-algebra $\mathcal{A}(F)$, and they prove that the Hochschild homology of this bimodule recovers the knot Floer homology of $K$. Therefore, one way to categorify Fox-Milnor would be to identify features of this bimodule when $K$ is slice which descend to the classical Fox-Milnor criterion after taking Hochschild homology and then the graded Euler characteristic. 

This approach is more plausible when $K$ is fibered, thanks to the remarkable theorem of Casson-Gordon which states that a fibered knot $K\subset S^3$ is homotopy-ribbon if and only if its closed monodromy extends over a handlebody. The closed monodromy $\phi$ of a fibered knot is the map induced by its monodromy on the capped-off fiber surface $F$. We will not worry about what homotopy-ribbon means except that it is implied by slice (the topological Slice-Ribbon conjecture is the assertion that the two notions are equivalent). 

When $K$ is fibered, the manifold $W$ above is just the mapping cylinder of $\phi$. That is, $$W=F\times [0,1]$$ is the bordered manifold where $F_0$ and $F_1$ are identified with $F$ by the identity and $\phi$, respectively. The goal is then to identify features of $\mathcal{F}_{DA}(W)$ implied by the condition that $\phi$ extends over a handlebody (the work of Lipshitz-Alishahi might be of use here), which descend to the classical Fox-Milnor criterion as above. There is a tractable first step here, suitable for an early graduate student:

Problem: Understand in terms of the decategorification of $\mathcal{F}_{DA}(W)$ why $\phi$ extending over a handlebody implies the Fox-Milnor criterion for the Alexander polynomial.

The bimodule $\mathcal{F}_{DA}(W)$ decategorifies to Donaldson's TQFT, as explained by Hom-Lidman-Watson (this is true in general, though we will assume that $W$ is the complement of a fiber surface as above). More concretely, let $g=g(F)$, and let $v_1,\dots,v_{2g}$ be a basis for the kernel $$\ker(i_*) \subset H_1(-F_0)\oplus H_1(F_1)$$ of the map induced by the inclusion $i:\partial W = -F_0\sqcup F_1 \to W.$ Define $$[W]:=v_1\wedge\dots\wedge v_{2g}\in \wedge^{2g}(H_1(-F_0)\oplus H_1(F_1)).$$ Since $\wedge^{2g}(H_1(-F_0)\oplus H_1(F_1))$ is contained in $$\bigoplus_{i=-g}^g \wedge^{g-i}(H_1(-F_0))\otimes \wedge^{g+i}(H_1(F_1))\cong \bigoplus_{i=-g}^g (\wedge^{g+i}(H_1(F_0)))^*\otimes \wedge^{g+i}(H_1(F_1)),$$ we can view $[W]$ as given by a family of maps $$\alpha_{W,j}:\wedge^{j}(H_1(F_0)) \to \wedge^{j}(H_1(F_1))$$ for $j = 0,\dots,2g$. Donaldson proves that the Alexander polynomial can be recovered from the traces of these maps, $$\sum_{j=0}^{2g}(-1)^j t^j\mathrm{Tr}(\alpha_{W,j}) = \Delta_K(t).$$ The goal of the tractable problem above is then to understand in terms of the maps $\alpha_{W,j}$ and this formula why the closed monodromy of $K$ extending over a handlebody implies that $\Delta_K(t)$ is of the form $f(t)f(t^{-1})$. Supposing $\phi$ fixes a handlebody $H$ whose attaching curves are given by $\alpha_1,\dots,\alpha_g\subset F$, one can show for example that $\alpha_{W,g}$ fixes the element $[H] = \alpha_1\wedge\dots\wedge\alpha_g$. 





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