arXiv subjects

Carmen Caprau

Publications and source records attributed to Carmen Caprau.

At least 19 recordsLinked to original sources

arXiv2026

On the structure of the singular triplet monoid and its virtual extension

In this article, we introduce two new algebraic structures associated with the triplet group on $n$ strands, $L_n$: the singular triplet monoid $SLM_n$ and its virtual extension $VSLM_n$, defined in analogy with the singular braid monoid and the virtual singular braid monoid. We begin by presenting these monoids in terms of generators and relations, and then derive several alternative presentations of $VSLM_n$. Second, we investigate the problem of extending representations of $L_n$ to these monoids. Two extension methods are developed: the $k$-local type extension, which applies to $k$-local representations, and the $\Phi$-type extension, which applies to representations satisfying suitable commutativity conditions. We show that every $2$-local representation of $L_n$ admits extensions to both $SLM_n$ and $VSLM_n$ via the two methods. As an application, we consider a specific representation $\mu : L_n \longrightarrow \mathrm{GL}_n(\mathbb{Z}[t^{\pm1}])$ introduced recently by Nasser et al. We explicitly determine all homogeneous $2$-local extensions of $\mu$ to $SLM_n$ and $VSLM_n$, and compute the corresponding $\Phi$-type extensions. Furthermore, we compare these two extension methods, showing that they coincide for $SLM_n$ under suitable parameter conditions, while they do not coincide for $VSLM_n$. These results provide a systematic framework for extending representations of $L_n$ to $SLM_n$ and $VSLM_n$.

math.RT↗

arXiv2026

An Extension of the $sl(n)$ Polynomial to Knotted 4-Valent Graphs

We use planar 4-valent graphs and a graphical calculus involving such graphs to construct an invariant for balanced-oriented, knotted 4-valent graphs. Our invariant is an extension of the $sl(n)$ polynomial for classical knots and links. We also provide a minimal generating set of Reidemeister-type moves for diagrams of balanced-oriented, knotted 4-valent graphs.

math.GT↗

arXiv2026

A whittled complex for the Khovanov homology of torus links

We give an algorithm for reducing the number of generators of the Khovanov chain complex of the torus braid $ft^k_n = (\sigma_1\sigma_2\cdots \sigma_{n-1})^k$ on $n$ strands by applying Bar-Natan Gaussian elimination along a distinguished set of Gaussian elimination isomorphisms. We call the resulting complex $\mathcal{FT}^k_n$ a \emph{whittled complex} for the Khovanov homology of torus braids. Using this algorithm, we provide a bound for the number of generators at a fixed homological degree in our whittled complex.

math.GT↗

arXiv2026

The virtual singular twin monoid and group: presentations and representations

In this article, we introduce the algebraic definitions and presentations of the virtual singular twin monoid and virtual singular twin group, denoted by $VSTM_n$ and $VST_n$, respectively, for a positive integer $n$. These structures extend the twin group $T_n$ in close analogy to how the virtual singular braid monoid and virtual singular braid group extend the classical braid group. We then construct and study representations of the group $VST_n$, for $n \geq 3$, focusing in particular on extending the representations $\eta_1$ and $\eta_2$ of $T_n$, introduced by M. Nasser, to $VST_n$ via the $2$-local extension method. To analyze the resulting representations, $\eta_1'$ and $\eta_2'$, and their properties, we establish necessary and sufficient conditions for irreducibility and show that both $\eta_1'$ and $\eta_2'$ are unfaithful. Additionally, we classify all complex homogeneous $2$-local representations of $VST_n$ for every integer $n\geq 3$, providing a foundation for further investigation into representations of $VST_n$.

math.RT↗

arXiv2024

Khovanov-Rozansky homology of Coxeter knots and Schröder polynomials for paths under any line

We introduce a family of generalized Schröder polynomials $S_τ(q,t,a)$, indexed by triangular partitions $τ$ and prove that $S_τ(q,t,a)$ agrees with the Poincaré series of the triply graded Khovanov-Rozansky homology of the Coxeter knot $K_τ$ associated to $τ$. For all integers $m,n,d\geq 1$ with $m,n$ relatively prime, the $(d,mnd+1)$-cable of the torus knot $T(m,n)$ appears as a special case. It is known that these knots are algebraic, and as a result we obtain a proof of the $q=1$ specialization of the Oblomkov-Rasmussen-Shende conjecture for these knots. Finally, we show that our Schröder polynomial computes the hook components in the Schur expansion of the symmetric function appearing in the shuffle theorem under any line, thus proving a triangular version of the $(q,t)$-Schröder theorem.

math.GT↗

arXiv2024

On an invariant for colored classical and singular links

A colored link, as defined by Francesca Aicardi, is an oriented classical link together with a coloration, which is a function defined on the set of link components and whose image is a finite set of colors. An oriented classical link can be regarded as a colored link with its components colored with a sole color. Aicardi constructed an invariant $F(L)$ of colored links $L$ defined via skein relations. When the components of a colored link are colored with the same color or when the colored link is a knot, $F(L)$ is a specialization of the HOMFLY-PT polynomial. Aicardi also showed that $F(L)$ is a stronger invariant than the HOMFLY-PT polynomial when evaluated on colored links whose components have different colors. In this paper, we provide a state-sum model for the invariant $F(L)$ of colored links using a graphical calculus for oriented, colored, 4-valent planar graphs. We also extend $F(L)$ to an invariant of oriented colored singular links.

math.GT↗

arXiv2022

Quandle Coloring Quivers of (p, 2)-Torus Links

A quandle coloring quiver is a quiver structure, introduced by Karina Cho and Sam Nelson, which is defined on the set of quandle colorings of an oriented knot or link by a finite quandle. We study quandle coloring quivers of (p, 2)-torus knots and links with respect to dihedral quandles.

math.GT↗

arXiv2022

Minimal generating sets of moves for diagrams of isotopic knots and spatial trivalent graphs

Polyak proved that all oriented versions of Reidemeister moves for knot and link diagrams can be generated by a set of just four oriented Reidemeister moves, and that no fewer than four oriented Reidemeister moves generate them all. We refer to a set containing four oriented Reidemeister moves that collectively generate all of the other oriented Reidemeister moves as a minimal generating set. Polyak also proved that a certain set containing two Reidemeister moves of type 1, one move of type 2, and one move of type 3 form a minimal generating set for all oriented Reidemeister moves. We expand upon Polyak's work by providing an additional eleven minimal, 4-element, generating sets of oriented Reidemeister moves, and we prove that these twelve sets represent all possible minimal generating sets of oriented Reidemeister moves. We also consider the Reidemeister-type moves that relate oriented spatial trivalent graph diagrams with trivalent vertices that are sources and sinks and prove that a minimal generating set of oriented Reidemeister-type moves for spatial trivalent graph diagrams contains ten moves.

math.GT↗

arXiv2022

Algebraic structures among virtual singular braids

We show that the virtual singular braid monoid on $n$ strands embeds in a group $VSG_n$, which we call the virtual singular braid group on $n$ strands. The group $VSG_n$ contains a normal subgroup $VSPG_n$ of virtual singular pure braids. We show that $VSG_n$ is a semi-direct product of $VSPG_n$ and the symmetric group $S_n$. We provide a presentation for $VSPG_n$ via generators and relations. We also represent $VSPG_n$ as a semi-direct product of $n-1$ subgroups and study the structures of these subgroups. These results yield a normal form of words in the virtual singular braid group.

math.GT↗

arXiv2020

On Khovanov Homology and Related Invariants

This paper begins with a survey of some applications of Khovanov homology to low-dimensional topology, with an eye toward extending these results to $\mathfrak{sl}(n)$ homologies. We extend Levine and Zemke's ribbon concordance obstruction from Khovanov homology to $\mathfrak{sl}(n)$ homology for $n \geq 2$, including the universal $\mathfrak{sl}(2)$ and $\mathfrak{sl}(3)$ homology theories. Inspired by Alishahi and Dowlin's bounds for the unknotting number coming from Khovanov homology and relying on spectral sequence arguments, we produce bounds on the alternation number of a knot. Lee and Bar-Natan spectral sequences also provide lower bounds on Turaev genus.

math.GT↗

arXiv2019

On the virtual singular braid monoid

We study the algebraic structures of the virtual singular braid monoid, $VSB_n$, and the virtual singular pure braid monoid, $VSP_n$. The monoid $VSB_n$ is the splittable extension of $VSP_n$ by the symmetric group $S_n$. We also construct a representation of $VSB_n$.

math.GT↗

arXiv2019

On the Kauffman-Jones polynomial for virtual singular links

We extend the Kamada-Miyazawa polynomial to virtual singular links, which is valued in $\mathbb{Z}[A^2, A^{-2}, h]$. The decomposition of the resulting polynomial into two components, one in $\mathbb{Z}[A^2, A^{-2}]$ and the other in $\mathbb{Z}[A^2, A^{-2}]h$ yields the decomposition of the Kauffman-Jones polynomial of virtual singular links into two components, one in $\mathbb{Z}[A^2, A^{-2}]$ and the other in $\mathbb{Z}[A^2, A^{-2}]A^2$, where both components are invariants for virtual singular links.

math.GT↗

arXiv2018

Invariants for trivalent tangles and handlebody-tangles

An enhanced trivalent tangle is a trivalent tangle with some of its edges labeled. We use enhanced trivalent tangles and classical knot theory to provide a recipe for constructing invariants for trivalent tangles, and in particular, for knotted trivalent graphs. Our method also yields invariants of, what we refer to as, enhanced handlebody-tangles and enhanced handlebody-links.

math.GT↗

arXiv2018

Alexander- and Markov-type theorems for virtual trivalent braids

We prove Alexander- and Markov-type theorems for virtual spatial trivalent graphs and virtual trivalent braids. We provide two versions for the Markov-type theorem: one uses an algebraic approach similar to the case of classical braids and the other one is based on L-moves.

math.GT↗

arXiv2018

The L-move and Markov theorems for trivalent braids

The L-move for classical braids extends naturally to trivalent braids. We follow the L-move approach to the Markov Theorem, to prove a one-move Markov-type theorem for trivalent braids. We also reformulate this L-Move Markov theorem and prove a more algebraic Markov-type theorem for trivalent braids. Along the way, we provide a proof of the Alexander's theorem analogue for spatial trivalent graphs and trivalent braids.

math.GT↗

arXiv2016

Virtual singular braids and links

Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer generators.

math.GT↗

arXiv2015

Movie moves for singular link cobordisms in 4-dimensional space

Two singular links are cobordant if one can be obtained from the other by singular link isotopy together with a combination of births or deaths of simple unknotted curves, and saddle point transformations. A movie description of a singular link cobordism in 4-space is a sequence of singular link diagrams obtained from a projection of the cobordism into 3-space by taking 2-dimensional cross sections perpendicular to a fixed direction. We present a set of movie moves that are sufficient to connect any two movies of isotopic singular link cobordisms.

math.GT↗

arXiv2015

Singular Links and Yang-Baxter State Models

We employ a solution of the Yang-Baxter equation to construct invariants for knot-like objects. Specifically, we consider a Yang-Baxter state model for the sl(n) polynomial of classical links and extend it to oriented singular links and balanced oriented 4-valent knotted graphs with rigid vertices. We also define a representation of the singular braid monoid into a matrix algebra, and seek conditions for extending further the invariant to contain topological knotted graphs. In addition, we show that the resulting Yang-Baxter-type invariant for singular links yields a version of the Murakami-Ohtsuki-Yamada state model for the sl(n) polynomial for classical links.

math.GT↗