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1. A note on cables and the involutive concordance invariants

   https://doi.org/10.1112/blms.70050  

   Hendricks, Kristen; Mallick, Abhishek (March 2025, Bulletin of the London Mathematical Society)

   Abstract We prove a formula for the involutive concordance invariants of the cabled knots in terms of those of the companion knot and the pattern knot. As a consequence, we show that any iterated cable of a knot with parameters of the form (odd,1) is not smoothly slice as long as either of the involutive concordance invariants of the knot is nonzero. Our formula also gives new bounds for the unknotting number of a cabled knot, which are sometimes stronger than other known bounds coming from knot Floer homology. 

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2. Exotic Mazur manifolds and knot trace invariants

   https://doi.org/10.1016/j.aim.2021.107994  

   Hayden, Kyle; Mark, Thomas E; Piccirillo, Lisa (November 2021, Advances in Mathematics)

   From a handle-theoretic perspective, the simplest contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance invariant ν is an invariant of the trace of a knot, i.e. the smooth 4-manifold obtained by attaching a 2-handle to the 4-ball along K. This provides a computable, integer-valued diffeomorphism invariant that is effective at distinguishing exotic smooth structures on knot traces and other simple 4-manifolds, including when other adjunction-type obstructions are ineffective. We also show that the concordance invariants τ and ϵ are not knot trace invariants. As a corollary to the existence of exotic Mazur manifolds, we produce integer homology 3-spheres admitting two distinct surgeries to $$S^1 \times S^2$$, resolving a question from Problem 1.16 in Kirby's list. 

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3. Equivariant knots and knot Floer homology

   https://doi.org/10.1112/topo.12312  

   Dai, Irving; Mallick, Abhishek; Stoffregen, Matthew (September 2023, Journal of Topology)

   Abstract We define several equivariant concordance invariants using knot Floer homology. We show that our invariants provide a lower bound for the equivariant slice genus and use this to give a family of strongly invertible slice knots whose equivariant slice genus grows arbitrarily large, answering a question of Boyle and Issa. We also apply our formalism to several seemingly nonequivariant questions. In particular, we show that knot Floer homology can be used to detect exotic pairs of slice disks, recovering an example due to Hayden, and extend a result due to Miller and Powell regarding stabilization distance. Our formalism suggests a possible route toward establishing the noncommutativity of the equivariant concordance group. 

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4. Stabilization distance bounds from link Floer homology

   https://doi.org/10.1112/topo.12338  

   Juhász, András; Zemke, Ian (June 2024, Journal of Topology)

   Abstract We consider the set of connected surfaces in the 4‐ball with boundary a fixed knot in the 3‐sphere. We define the stabilization distance between two surfaces as the minimal such that we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most . Similarly, we consider a double‐point distance between two surfaces of the same genus that is the minimum over all regular homotopies connecting the two surfaces of the maximal number of double points appearing in the homotopy. To many of the concordance invariants defined using Heegaard Floer homology, we construct an analogous invariant for a pair of surfaces. We show that these give lower bounds on the stabilization distance and the double‐point distance. We compute our invariants for some pairs of deform‐spun slice disks by proving a trace formula on the full infinity knot Floer complex, and by determining the action on knot Floer homology of an automorphism of the connected sum of a knot with itself that swaps the two summands. We use our invariants to find pairs of slice disks with arbitrarily large distance with respect to many of the metrics we consider in this paper. We also answer a slice‐disk analog of Problem 1.105 (B) from Kirby's problem list by showing the existence of non‐0‐cobordant slice disks. 

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5. The ϒ function of L –space knots is a Legendre transform

   https://doi.org/10.1017/S030500411700024X  

   BORODZIK, MACIEJ; HEDDEN, MATTHEW (May 2018, Mathematical Proceedings of the Cambridge Philosophical Society)

   null (Ed.)

   Abstract Given an L –space knot we show that its ϒ function is the Legendre transform of a counting function equivalent to the d –invariants of its large surgeries. The unknotting obstruction obtained for the ϒ function is, in the case of L –space knots, contained in the d –invariants of large surgeries. Generalisations apply for connected sums of L –space knots, which imply that the slice obstruction provided by ϒ on the subgroup of concordance generated by L –space knots is no finer than that provided by the d –invariants. 

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