The unknotting number of 11n102 is 2
We prove that the unknotting number of the knot 11n102 is 2.
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arXiv subjects
Tye Lidman
Publications and source records attributed to Tye Lidman.
A homological generalized Property R conjecture is false
The generalized Property R conjecture (GPRC) predicts that if framed surgery on an $n$-component link $L$ in $S^3$ produces $\#^{n} (S^1\times S^2)$, then $L$ is handleslide equivalent to an unlink, the obvious way to construct such a surgery. Many potential counterexamples to the GPRC are known, but obstructing handleslide equivalence is a tricky proposition. In this vein, we disprove a further generalization of the GPRC. It would be reasonable to expect that if an $n$-component link in $S^3$ surgers to the connected sum of $n$ three-manifolds with the homology of $S^1 \times S^2$, then this link should be handleslide equivalent to an $n$-component split link, the obvious way to construct such a surgery. However, we prove that there are 2-component framed links in $S^3$ that surger to a connected sum of homology $S^1\times S^2$'s but that are not handleslide equivalent, or even weakly handleslide equivalent, to a split link.
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Distinguishing exotic $\mathbb{R}^4$'s with Heegaard Floer homology
Attaching a Casson handle to a slice disk complement yields a smooth 4-manifold that is homeomorphic to $\mathbb{R}^4$. We show that if two slice knots have sufficiently different knot Floer homology, then the resulting exotic $\mathbb{R}^4$'s made using the simplest positive Casson handle are not diffeomorphic, giving us a countably infinite family of pairwise nondiffeomorphic chiral exotic $\mathbb{R}^4$'s. Our main tool is Gadgil's end Floer homology and we use this to produce families of exotic $\mathbb{R}^4$ with various phenomena. As an application, we reprove a result of Bi\v{z}aca-Etnyre that $Y \times \mathbb{R}$, where $Y$ is any closed $3$-manifold, has infinitely many distinct smooth structures.
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Homology cobordism for some Sol manifolds
We show that two 3-manifolds that admit Sol geometry and have first homology group of order 16 are integer homology cobordant if and only if they are homeomorphic.
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The knot complement problem for null-homotopic knots
We prove that for three-manifolds satisfying a certain algebraic condition on their fundamental group, null-homotopic knots are determined by their complements. This answers a Kirby Problem posed by Boileau for this special case of 3-manifolds. The argument uses techniques in instanton Floer homology and SU(2)-representation varieties.
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Not all knots are smoothly round handle slice
Freedman and Krushkal showed that if the surgery conjecture and the $s$-cobordism conjecture hold for all topological 4-manifolds, then every link with pairwise zero linking numbers is topologically round handle slice. Kim, Powell, and Teichner showed that every knot is topologically round handle slice. We show that infinitely many knots fail to be smoothly round handle slice.
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Distinguishing closed 4-manifolds by slicing
One approach to produce a pair of homeomorphic-but-not-diffeomophic closed 4-manifolds is to find a knot which is smoothly slice in one but not the other. This approach has never been run successfully. We give the first examples of a pair of closed 4-manifolds with the same integer cohomology ring where the diffeomorphism type is distinguished by this approach. Along the way, we produce the first examples of 4-manifolds with nonvanishing Seiberg-Witten invariants and the same integer cohomology as $\mathbb{C}P^2\#\overline{\mathbb{C}P^2}$ which are not diffeomorphic to $\mathbb{C}P^2\#\overline{\mathbb{C}P^2}$. We also give a simple new construction of a 4-manifold which is homeomorphic-but-not-diffeomorphic to $\mathbb{C}P^2\#5\overline{\mathbb{C}P^2}$.
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Detecting Heegaard Floer homology solid tori
We show that a rational homology solid torus is a Heegaard Floer homology solid torus if and only if it has a Dehn filling with a non-separating 2-sphere. Using this, we characterize Seifert fibered Heegaard Floer solid tori.
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Stably irreducible non-orientable knotted surfaces
We give an elementary obstruction to reducibility for knotted surfaces in the four-sphere. As a new application, we construct stably irreducible non-orientable surfaces.
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Heegaard Floer homology and the word metric on the Torelli group
We study a relationship between the Heegaard Floer homology correction terms of integral homology spheres and the word metric on the Torelli group. For example, we give an elementary proof that the Cayley graph of the Torelli group has infinite diameter in the word metric induced by the generating set of all separating twists and bounding pair maps. On the other hand, we show that many subsets of the Torelli group are bounded with respect to this metric. Finally, we address the case of rational homology spheres by ruling out a certain Morita-type formula for congruence subgroups of mapping class groups.
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Filtered instanton homology and cosmetic surgery
The cosmetic surgery conjecture predicts that for a non-trivial knot in the three-sphere, performing two different Dehn surgeries results in distinct oriented three-manifolds. Hanselman reduced the problem to $\pm 2$ or $\pm 1/n$ surgeries being the only possible cosmetic surgeries. We remove the case of $\pm 1/n$-surgeries using the Chern-Simons filtration on Floer's original irreducible-only instanton homology, reducing the conjecture to the case of $\pm 2$ surgery on genus $2$ knots with trivial Alexander polynomial. We also prove some similar results for surgeries on knots in $S^2 \times S^1$. As key steps in establishing these results, we define invariants of the oriented homeomorphism type of three-manifolds derived from filtered instanton Floer homology and introduce a new surgery relationship for Floer's instanton homology.
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Spineless 5-manifolds and the deformation conjecture
We construct a compact PL 5-manifold $M$ (with boundary) which is homotopy equivalent to the wedge of eleven 2-spheres, $\vee^{}_{1 1}S^2$, which is "spineless", meaning $M$ is not the regular neighborhood of any 2-complex PL embedded in $M$. We formulate a related question about the existence of exotic smooth structures on 4-manifolds which is of interest in relation to the deformation conjecture for 2-complexes, also known as the generalized Andrews-Curtis conjecture.
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Adjacency of three-manifolds and Brunnian links
We introduce the notion of adjacency in three-manifolds. A three-manifold $Y$ is $n$-adjacent to another three-manifold $Z$ if there exists an $n$-component link in $Y$ and surgery slopes for that link such that performing Dehn surgery along any nonempty sublink yields $Z$. We characterize adjacencies from three-manifolds to the three-sphere, providing an analogy to Askitas and Kalfagianni's results on $n$-adjacency in knots.
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New constructions and invariants of closed exotic 4-manifolds
In this article, we give new means of constructing and distinguishing closed exotic four-manifolds. Using Heegaard Floer homology, we define new closed four-manifold invariants that are distinct from the Seiberg--Witten and Bauer--Furuta invariants and can remain distinct in covers. Our constructions include exotic definite manifolds with fundamental group $\mathbb Z/2$, infinite families of exotic manifolds that are related by knot surgeries on Alexander polynomial 1 knots, and exotic manifolds that contain square-zero spheres.
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Knot concordance in homology cobordisms
Let $\widehat{\mathcal{C}}_{\mathbb{Z}}$ denote the group of knots in homology spheres that bound homology balls, modulo smooth concordance in homology cobordisms. Answering a question of Matsumoto, the second author previously showed that the natural map from the smooth knot concordance group $\mathcal{C}$ to $\widehat{\mathcal{C}}_{\mathbb{Z}}$ is not surjective. Using tools from Heegaard Floer homology, we show that the cokernel of this map, which can be understood as the non-locally-flat piecewise-linear concordance group, is infinitely generated and contains elements of infinite order.
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Handle decomposition complexity and representation spaces
We prove that there are homology three-spheres that bound definite four-manifolds, but any such bounding four-manifold must be built out of many handles. The argument uses the homology cobordism invariant $\Gamma$ from instanton Floer homology.
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Linking number obstructions to satellite homomorphisms
We prove that satellite operations that satisfy a certain positivity condition and have winding number other than one are not homomorphisms. The argument uses the $d$-invariants of branched covers. In the process, we prove a technical result relating $d$-invariants and the Torelli group which may be of independent interest.
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3-manifolds without any embedding in symplectic 4-manifolds
We show that there exist infinitely many closed 3-manifolds that do not embed in closed symplectic 4-manifolds, disproving a conjecture of Etnyre-Min-Mukherjee. To do this, we construct L-spaces that cannot bound positive or negative definite manifolds. The arguments use Heegaard Floer correction terms and instanton moduli spaces.
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