arXiv subjects
Publications and source records attributed to Lisa Piccirillo.
We prove that, for each fixed rational number $p/q \in \mathbb{Q}$, there exists a pair of distinct knots whose $p/q$-surgeries are orientation-preservingly homeomorphic. This confirms a 1978 conjecture of Gordon.
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 $\mathbb{C}P^2\#5\overline{\mathbb{C}P^2}$.
Distinct knots K, K' can sometimes share a common p/q-framed Dehn surgery. A folk conjecture held that for a fixed pair of knots, this can occur for at most one value of p/q. We disprove this conjecture by constructing pairs of distinct knots K,K' that have common Dehn surgeries for four distinct slopes. We also construct non-isotopic Legendrian knots K,K' that have contactomorphic contact (+1)-and (-1)-surgeries, disproving an analogous conjecture in contact geometry.
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.
We provide new examples of 3-manifolds with weight one fundamental group and the same integral homology as the lens space $L(2k,1)$ which are not surgery on any knot in the three-sphere. Our argument uses Furuta's 10/8-theorem, and is simple and combinatorial to apply.
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.
We present a handlebody construction of small symplectic caps, and hence of small closed symplectic 4-manifolds. We use this to construct handlebody descriptions of symplectic embeddings of rational homology balls in $\mathbb{C}\mathrm{P}^2$, and thereby provide the first examples of (infinitely many) symplectic handlebody decompositions of a closed symplectic 4-manifold. Our constructions provide a new topological interpretation of almost toric fibrations of $\mathbb{C}\mathrm{P}^2$ in terms of symplectic handlebody decompositions.
We classify topological $4$-manifolds with boundary and fundamental group $\mathbb{Z}$, under some assumptions on the boundary. We apply this to classify surfaces in simply-connected $4$-manifolds with $S^3$ boundary, where the fundamental group of the surface complement is $\mathbb{Z}$. We then compare these homeomorphism classifications with the smooth setting. For manifolds, we show that every Hermitian form over $\mathbb{Z}[t^{\pm 1}]$ arises as the equivariant intersection form of a pair of exotic smooth 4-manifolds with boundary and fundamental group $\mathbb{Z}$. For surfaces we have a similar result, and in particular we show that every $2$-handlebody with $S^3$ boundary contains a pair of exotic discs.
One strategy for distinguishing smooth structures on closed $4$-manifolds is to produce a knot $K$ in $S^3$ that is slice in one smooth filling $W$ of $S^3$ but not slice in some homeomorphic smooth filling $W'$. In this paper we explore how $0$-surgery homeomorphisms can be used to potentially construct exotic pairs of this form. In order to systematically generate a plethora of candidates for exotic pairs, we give a fully general construction of pairs of knots with the same zero surgeries. By computer experimentation, we find $5$ topologically slice knots such that, if any of them were slice, we would obtain an exotic four-sphere. We also investigate the possibility of constructing exotic smooth structures on $\#^n \mathbb{C}P^2$ in a similar fashion.
Given a closed four-manifold $X$ with an indefinite intersection form, we consider smoothly embedded surfaces in $X \setminus $int$(B^4)$, with boundary a knot $K \subset S^3$. We give several methods to bound the genus of such surfaces in a fixed homology class. Our tools include adjunction inequalities and the $10/8 + 4$ theorem. In particular, we present obstructions to a knot being H-slice (that is, bounding a null-homologous disk) in a four-manifold and show that the set of H-slice knots can detect exotic smooth structures on closed $4$-manifolds.
A cork is a smooth, contractible, oriented, compact 4-manifold $W$ together with a self-diffeomorphism $f$ of the boundary 3-manifold that cannot extend to a self-diffeomorphism of $W$; the cork is said to be strong if $f$ cannot extend to a self-diffeomorphism of any smooth integer homology ball bounded by $\partial W$. Surprising recent work of Dai, Hedden, and Mallick showed that most of the well-known corks in the literature are strong. We construct the first non-strong corks, which also give rise to new examples of absolutely exotic Mazur manifolds. Additionally we give the first examples of corks where the diffeomorphism of $\partial W$ can be taken to be orientation-reversing.
We demonstrate new applications of the trace embedding lemma to the study of piecewise-linear surfaces and the detection of exotic phenomena in dimension four. We provide infinitely many pairs of homeomorphic 4-manifolds $W$ and $W'$ homotopy equivalent to $S^2$ which have smooth structures distinguished by several formal properties: $W'$ is diffeomorphic to a knot trace but $W$ is not, $W'$ contains $S^2$ as a smooth spine but $W$ does not even contain $S^2$ as a piecewise-linear spine, $W'$ is geometrically simply connected but $W$ is not, and $W'$ does not admit a Stein structure but $W$ does. In particular, the simple spineless 4-manifolds $W$ provide an alternative to Levine and Lidman's recent solution to Problem 4.25 in Kirby's list. We also show that all smooth 4-manifolds contain topological locally flat surfaces that cannot be approximated by piecewise-linear surfaces.
From a handlebody-theoretic perspective, the simplest compact, 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 smooth 4-manifold associated to a knot in the 3-sphere by attaching an n-framed 2-handle to the 4-ball along the knot. In contrast, we also show (modulo forthcoming work of Ozsváth and Szabó) that the concordance invariants $τ$ and $ε$ are not invariants of such 4-manifolds. As a corollary to the existence of exotic Mazur manifolds, we produce integer homology 3-spheres admitting two distinct $S^1 \times S^2$ surgeries, resolving a question from Problem 1.16 in Kirby's list.
An important difference between high dimensional smooth manifolds and smooth 4-manifolds that in a 4-manifold it is not always possible to represent every middle dimensional homology class with a smoothly embedded sphere. This is true even among the simplest 4-manifolds: $X_0(K)$ obtained by attaching an $0$-framed 2-handle to the 4-ball along a knot $K$ in $S^3$. The $0$-shake genus of $K$ records the minimal genus among all smooth embedded surfaces representing a generator of the second homology of $X_0(K)$ and is clearly bounded above by the slice genus of $K$. We prove that slice genus is not an invariant of $X_0(K)$, and thereby provide infinitely many examples of knots with $0$-shake genus strictly less than slice genus. This resolves Problem 1.41 of [Kir97]. As corollaries we show that Rasmussen's $s$ invariant is not a $0$-trace invariant and we give examples, via the satellite operation, of bijective maps on the smooth concordance group which fix the identity but do not preserve slice genus. These corollaries resolve some questions from [4MKC16].
A knot is said to be slice if it bounds a smooth properly embedded disk in the 4-ball. We demonstrate that the Conway knot, 11n34 in the Rolfsen tables, is not slice. This completes the classification of slice knots under 13 crossings, and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.
We give a method for constructing many pairs of distinct knots $K_0$ and $K_1$ such that the two 4-manifolds obtained by attaching a 2-handle to $B^4$ along $K_i$ with framing zero are diffeomorphic. We use the d-invariants of Heegaard Floer homology to obstruct the smooth concordance of some of these $K_0$ and $K_1$, thereby disproving a conjecture of Abe in [Abe16]. As a consequence, we obtain a proof that there exist patterns $P$ in solid tori such that $P(K)$ is not always concordant to $P(U) \# K$ and yet whose action on the smooth concordance group is invertible.
For an integer $n$, write $X_n(K)$ for the 4-manifold obtained by attaching a 2-handle to the 4-ball along the knot $K\subset S^3$ with framing $n$. It is known that if $n< \overline{\text{tb}}(K)$, then $X_n(K)$ admits the structure of a Stein domain, and moreover the adjunction inequality implies there is an upper bound on the value of $n$ such that $X_n(K)$ is Stein. We provide examples of knots $K$ and integers $n\geq \overline{\text{tb}}(K)$ for which $X_n(K)$ is Stein, answering an open question in the field. In fact, our family of examples shows that the largest framing such that the manifold $X_n(K)$ admits a Stein structure can be arbitrarily larger than $\overline{\text{tb}}(K)$. We also provide an upper bound on the Stein framings for $K$ that is typically stronger than that coming from the adjunction inequality.