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Paolo Lisca

Publications and source records attributed to Paolo Lisca.

At least 19 recordsLinked to original sources
arXiv2026

A Polynomial Invariant of Strongly Involutive Links

We introduce a new two-variable polynomial invariant \(P^e\) of strongly involutive links, uniquely characterised by equivariant skein relations and naturally viewed as an equivariant analogue of the HOMFLY--PT polynomial. We prove that a specialisation of \(P^e\) recovers the graded Euler characteristic of the third page of the Lobb--Watson \(\mathcal{G}\)-filtration spectral sequence, generalising Couture's polynomial invariant. We further show that, after a change of variables, \(P^e\) reduces modulo \(2\) to the HOMFLY--PT polynomial, up to an explicit power of the skein variable, thereby answering a generalized form of a question of Couture. We use the resulting skein relations to distinguish infinitely many pairs of alternating mutant knots, and show that \(P^e\) is strictly stronger than the refined Lobb--Watson invariants on infinitely many strongly invertible knots.

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arXiv2023

Strongly Invertible Legendrian Links

We introduce and study strongly invertible Legendrian links in the standard contact three-dimensional space. We establish the equivariant analogs of basic results separately well-known for strongly invertible and Legendrian links, i.e. the existence of transvergent front diagrams, an equivariant Legendrian Reidemeister theorem, and an equivariant stabilization theorem à la Fuch-Tabachnikov. We also introduce a maximal equivariant Thurston-Bennequin number for strongly invertible links and we exhibit infinitely many such links for which the invariant coincides with the usual maximal Thurston-Bennequin number. We conjecture that such a coincidence does not hold general and that there exist strongly invertible knots having Legendrian representatives isotopic to their reversed Legendrian mirrors but not isotopic to any strongly invertible Legendrian knot.

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arXiv2023

On almost complex embeddings of rational homology balls

We use elementary arguments to prove that none of the Stein rational homology 4-balls shown by the authors and Brendan Owens to embed smoothly but not symplectically in the complex projective plane admit such almost complex embeddings. In particular, we are able to show that those rational balls admit no symplectic embeddings in the complex projective plane without appealing to the work of Evans-Smith.

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arXiv2023

Horizontal decompositions, II

We complete the classification of the smooth, closed, oriented 4-manifolds having Euler characteristic less than four and a horizontal handlebody decomposition of genus one. We use the classification result to find a large family of rational homology ball smoothings of cyclic quotient singularities which can be smoothly embedded into the complex projective plane. Our family contains all such rational balls previously known to embed into $\mathbb{CP}^2$ and infinitely many more. We also show that a rational ball of our family admits an almost-complex embedding in $\mathbb{CP}^2$ if and only if it admits a symplectic embedding.

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arXiv2022

Horizontal decompositions, I

We show that every smooth, closed, orientable 4-manifold X admits a special kind of handlebody decomposition that we call horizontal. We classify the closed 4-manifolds with the simplest horizontal decompositions and we describe all such decompositions of CP^2, showing that they give rise to infinitely many of the known embeddings of rational homology balls in the complex projective plane.

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arXiv2019

On symmetric equivalence of symmetric union diagrams

Eisermann and Lamm introduced a notion of symmetric equivalence among symmetric union diagrams and studied it using a refined form of the Jones polynomial. We introduced invariants of symmetric equivalence via refined versions of topological spin models and provided a partial answer to a question left open by Eisermann and Lamm. In this paper we adopt a new approach to the symmetric equivalence problem and give a complete answer to the original question left open by Eisermann and Lamm.

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arXiv2018

Symmetric union diagrams and refined spin models

An open question akin to the slice-ribbon conjecture asks whether every ribbon knot can be represented as a symmetric union. Next to this basic existence question sits the question of uniqueness of such representations. Eisermann and Lamm investigated the latter question by introducing a notion of symmetric equivalence among symmetric union diagrams and showing that inequivalent diagrams can be detected using a refined version of the Jones polynomial. We prove that every topological spin model gives rise to many effective invariants of symmetric equivalence, which can be used to distinguish infinitely many symmetric union diagrams representing the same link. We also show that such invariants are distinct from the refined Jones polynomial and we use them to provide a partial answer to a question left open by Eisermann and Lamm.

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arXiv2018

Framing 3-manifolds with bare hands

After surveying existing proofs that every closed, orientable 3-manifold is parallelizable, we give three proofs using minimal background. In particular, our proofs use neither spin structures nor the theory of Stiefel-Whitney classes.

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arXiv2015

On Stein fillings of contact torus bundles

We consider a large family F of torus bundles over the circle, and we use recent work of Li--Mak to construct, on each Y in F, a Stein fillable contact structure C. We prove that (i) each Stein filling of (Y,C) has vanishing first Chern class and first Betti number, (ii) if Y in F is elliptic then all Stein fillings of (Y,C) are pairwise diffeomorphic and (iii) if Y in F is parabolic or hyperbolic then all Stein fillings of (Y,C) share the same Betti numbers and fall into finitely many diffeomorphism classes. Moreover, for infinitely many hyperbolic torus bundles Y in F we exhibit non-homotopy equivalent Stein fillings of (Y,C).

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arXiv2014

Open book decompositions versus prime factorizations of closed, oriented 3-manifolds

Let $M$ be a closed, oriented, connected 3--manifold and $(B,π)$ an open book decomposition on $M$ with page $Σ$ and monodromy $φ$. It is easy to see that the first Betti number of $Σ$ is bounded below by the number of $S^2\times S^1$--factors in the prime factorization of $M$. Our main result is that equality is realized if and only if $φ$ is trivial and $M$ is a connected sum of $S^2\times S^1$'s. We also give some applications of our main result, such as a new proof of the result by Birman and Menasco that if the closure of a braid with $n$ strands is the unlink with $n$ components then the braid is trivial.

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arXiv2013

Signatures, Heegaard Floer correction terms and quasi-alternating links

Turaev showed that there is a well-defined map assigning to an oriented link L in the three-sphere a Spin structure t_0 on Sigma(L), the 2-fold cover of S^3 branched along L. We prove, generalizing results of Manolescu-Owens and Donald-Owens, that for an oriented quasi-alternating link L the signature of L equals minus four times the Heegaard Floer correction term of (Sigma(L), t_0).

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arXiv2013

Stein fillable contact 3-manifolds and positive open books of genus one

A two-dimensional open book (S,h) determines a closed, oriented three-manifold Y(S,h) and a contact structure C(S,h) on Y(S,h). The contact structure C(S,h) is Stein fillable if h is positive, i.e. h can be written as a product of right-handed Dehn twists. Work of Wendl implies that when S has genus zero the converse statement holds, that is if C(S,h) is Stein fillable then h is positive. On the other hand, results by Wand and by Baker, Etnyre and Van Horn-Morris imply the existence of counterexamples to the converse statement with S of arbitrary genus strictly greater than one. The main purpose of this paper is to prove the converse statement under the assumption that S is a one-holed torus and Y(S,h) is a Heegaard Floer L-space.

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arXiv2012

On overtwisted, right-veering open books

We exhibit infinitely many overtwisted, right-veering, non-destabilizable open books, thus providing infinitely many counterexamples to a conjecture of Honda-Kazez-Matic. The page of all our open books is a four-holed sphere and the underlying 3-manifolds are lens spaces.

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arXiv2011

Contact surgery and transverse invariants

We derive new existence results for tight contact structures on certain 3-manifolds which can be presented as surgery along specific knots in S^3. Indeed, we extend our earlier results on knots with maximal Thurston-Bennequin number being equal to 2g-1 to knots for which the maximal self-linking number satisfies the same equality. In the argument (using contact surgery) we define an invariant for transverse knots in contact 3-manifolds under the assumption that either the knot is null-homologous or the 3-manifold has no S^1xS^2-factor in its prime decomposition, and we study its properties using the Ozsvath-Szabo contact invariant.

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arXiv2010

Stein fillable Seifert fibered 3-manifolds

We characterize the closed, oriented, Seifert fibered 3-manifolds which are oriented boundaries of Stein manifolds. We also show that for this class of 3-manifolds the existence of Stein fillings is equivalent to the existence of symplectic fillings.

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arXiv2009

Heegaard Floer invariants of Legendrian knots in contact three--manifolds

We define invariants of null--homologous Legendrian and transverse knots in contact 3--manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non--loose knots in overtwisted 3--spheres. Moreover, we apply the invariants to find transversely non--simple knot types in many overtwisted contact 3--manifolds.

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