arXiv subjects

Carlo Collari

Publications and source records attributed to Carlo Collari.

22 records · Page 2Linked to original sources

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.

math.GT↗

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.

math.GT↗

arXiv2018

A Bennequin-type inequality and combinatorial bounds

In this paper we provide a new Bennequin-type inequality for the Rasmussen- Beliakova-Wehrli invariant, featuring the numerical transverse braid invariants (the c-invariants) introduced by the author. From the Bennequin type-inequality, and a combinatorial bound on the value of the c-invariants, we deduce a new computable bound on the Rasmussen invariant.

math.GT↗

arXiv2018

On transverse invariants from Khovanov-type homologies

In this article we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan's deformation. These invariants, called $β$-invariants, are essentially equivalent to Lipshitz, Ng, and Sarkar's invariants $ψ^\pm$. From the $β$-invariants we extract two non-negative integers which are transverse invariants (the $c$-invariants). Finally, we give several conditions which imply the non-effectiveness of the $c$-invariants, and use them to prove several vanishing criteria for the Plamenevskaya invariant $[ψ]$, and the non-effectiveness of the vanishing of $[ψ]$, for all prime knots with less than 12 crossings.

math.GT↗