6 citations · 28 across the 20 of their papers we have counts for
22 papers
A Polynomial Invariant of Strongly Involutive Links
Carlo Collari, Paolo Lisca
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 equivari…
Multipath complexes of bidirectional polygonal digraphs
Luigi Caputi, Carlo Collari, Jason P. Smith
In this work we study the homotopy type of multipath complexes of bidirectional path graphs and polygons, motivated by works of Vrećica and Živaljević on cycle-free chessboard comp…
Multipath matroids, digraph colourings, and the Tutte polynomial
Luigi Caputi, Carlo Collari, Sabino Di Trani
We characterise the digraphs for which the multipaths, that is disjoint unions of directed paths, yield a matroid. For such graphs, called MP-digraphs, we prove that the Tutte poly…
Bridging between überhomology and double homology
Luigi Caputi, Daniele Celoria, Carlo Collari
We establish an isomorphism between the 0-degree überhomology and the double homology of finite simplicial complexes, using a Mayer-Vietoris spectral sequence argument. We clarify…
The weak categorical quiver minor theorem and its applications: matchings, multipaths, and magnitude cohomology
Luigi Caputi, Carlo Collari, Eric Ramos
Building upon previous works of Proudfoot and Ramos, and using the categorical framework of Sam and Snowden, we extend the weak categorical minor theorem from undirected graphs to…
Strongly Invertible Legendrian Links
Carlo Collari, Paolo Lisca
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-k…
From the Mayer-Vietoris spectral sequence to überhomology
Luigi Caputi, Daniele Celoria, Carlo Collari
We prove that the second page of the Mayer-Vietoris spectral sequence, with respect to anti-star covers, can be identified with another homological invariant of simplicial complexe…
On finite generation in magnitude (co)homology, and its torsion
Luigi Caputi, Carlo Collari
The aim of this paper is to apply the framework, which was developed by Sam and Snowden, to study structural properties of graph homologies, in the spirit of Ramos, Miyata and Prou…
A note on weight systems which are quantum states
Carlo Collari
A result of Corfield, Sati, and Schreiber asserts that -weight systems associated to the defining representation are quantum states. In this short note we extend t…
On the homotopy type of multipath complexes
Luigi Caputi, Carlo Collari, Sabino Di Trani +1
A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph is the simplicial complex whose faces are the multipaths of ${\tt…
Monotone cohomologies and oriented matchings
Luigi Caputi, Daniele Celoria, Carlo Collari
In this paper, we extend the definition of cohomology associated to monotone graph properties, to encompass twisted functor coefficients. We introduce oriented matchings on graphs,…
Categorifying connected domination via graph überhomology
Luigi Caputi, Daniele Celoria, Carlo Collari
Überhomology is a recently defined homology theory for simplicial complexes, which yields subtle information on graphs. We prove that bold homology, a certain specialisation of übe…
Combinatorial and Topological Aspects of Path Posets, and Multipath Cohomology
Luigi Caputi, Carlo Collari, Sabino Di Trani
Multipath cohomology is a cohomology theory for directed graphs, which is defined using the path poset. The aim of this paper is to investigate combinatorial properties of path pos…
Multipath cohomology of directed graphs
Luigi Caputi, Carlo Collari, Sabino Di Trani
This work is part of a series of papers focusing on multipath cohomology of directed graphs. Multipath cohomology is defined as the (poset) homology of the path poset -- i.e., the…
Slice-torus link invariants, combinatorial invariants, and positivity conditions
Carlo Collari
We prove some necessary conditions for a link to be either concordant to a quasi-positive link, quasi-positive, positive, or the closure of a positive braid. The main applications…
A note on the weak splitting number
Alberto Cavallo, Carlo Collari, Anthony Conway
The weak splitting number of a link is the minimal number of crossing changes needed to turn into a split union of knots. We describe conditions under which certai…
On symmetric equivalence of symmetric union diagrams
Carlo Collari, Paolo Lisca
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…
Slice-torus concordance invariants and Whitehead doubles of links
Alberto Cavallo, Carlo Collari
In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under cro…
Transverse link invariants from the deformations of Khovanov -homology
Carlo Collari
In this paper we will make use of the Mackaay-Vaz approach to the universal -homology to define a family of cycles (called -invariants) which are transverse b…
Symmetric union diagrams and refined spin models
Carlo Collari, Paolo Lisca
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…
A Bennequin-type inequality and combinatorial bounds
Carlo Collari
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) intro…
On transverse invariants from Khovanov-type homologies
Carlo Collari
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 a…