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Lorenzo Traldi

Publications and source records attributed to Lorenzo Traldi.

At least 19 recordsLinked to original sources
arXiv2026

A note on longitudes of virtual knots

It is a famous property that a longitude of a classical knot lies in the second commutator subgroup of the knot group. We observe that the same property holds for a longitude of a virtual knot.

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arXiv2026

Peripheral structures of core groups

The core group is an invariant of unoriented virtual links. We introduce a peripheral structure for the core group, in which the longitudes are sensitive to orientations. We show that the combination of the core group and its peripheral structure is equivalent, as a link invariant, to the combination of the $π$-orbifold group and its peripheral structure. Examples show that the peripheral structure of the core group can be used to verify noninvertibility of some knots and links.

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arXiv2024

Reorienting quandle orbits

Motivated by knot theory, it is natural to define the orientation-reversal of a quandle orbit by inverting all the translations given by elements of that orbit. In this short note we observe that this natural notion is unsuited to medial quandles.

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arXiv2024

Multivariate Alexander quandles, VI. Metabelian groups and 2-component links

We prove two properties of the modules and quandles discussed in this series. First, the fundamental multivariate Alexander quandle $Q_A(L)$ is isomorphic to the natural image of the fundamental quandle in the metabelian quotient $G(L)/G(L)''$ of the link group. Second, the medial quandle of a classical 2-component link $L$ is determined by the reduced Alexander invariant of $L$.

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arXiv2024

Core groups

The core group of a classical link was introduced independently by A.J. Kelly in 1991 and M. Wada in 1992. It is a link invariant defined by a presentation involving the arcs and crossings of a diagram, related to Wirtinger's presentation of the fundamental group of a link complement. Two close relatives of the core group are defined by presentations involving regions rather than arcs; one of them is related to Dehn's presentation of a link group. The definitions are extended to virtual link diagrams and properties of the resulting invariants are discussed.

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arXiv2024

Circuit partitions and signed interlacement in 4-regular graphs

Let $F$ be a 4-regular graph. Each circuit partition $P$ of $F$ has a corresponding touch-graph $Tch(P)$; the circuits in $P$ correspond to vertices of $Tch(P)$, and the vertices of $F$ correspond to edges of $Tch(P)$. We discuss the connection between modified versions of the interlacement matrix of an Euler system of $F$ and the cycle space of $Tch(P)$, over $GF(2)$ and $\mathbb{R}$.

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arXiv2022

Peripheral elements in reduced Alexander modules

We discuss meridians and longitudes in reduced Alexander modules of classical and virtual links. When these elements are suitably defined, each link component will have many meridians, but only one longitude. Enhancing the reduced Alexander module by singling out these peripheral elements provides a significantly stronger link invariant. In particular, the enhanced module determines all linking numbers in a link; in contrast, the module alone does not even detect how many linking numbers are $0$.

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arXiv2022

A note on geometric duality in matroid theory and knot theory

We observe that for planar graphs, the geometric duality relation generates both 2-isomorphism and abstract duality. This observation has the surprising consequence that for links, the equivalence relation defined by isomorphisms of checkerboard graphs is the same as the equivalence relation defined by 2-isomorphisms of checkerboard graphs.

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arXiv2021

Multivariate Alexander quandles, IV. The medial quandle of a link

Joyce observed that the Alexander invariant and the medial quandle of a classical knot are equivalent to each other, as invariants. In the present paper, we discuss the rather complicated extension of Joyce's observation to several different medial quandles and reduced (one-variable) Alexander modules associated with classical links. The theme is that for links, medial quandles provide stronger invariants than reduced Alexander modules.

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arXiv2021

A Characterization of Circle Graphs in Terms of Total Unimodularity

A graph $G$ has an associated multimatroid $\mathcal{Z}_3(G)$, which is equivalent to the isotropic system of $G$ studied by Bouchet. In previous work it was shown that $G$ is a circle graph if and only if for every field $\mathbb F$, the rank function of $\mathcal{Z}_3(G)$ can be extended to the rank function of an $\mathbb F$-representable matroid. In the present paper we strengthen this result using a multimatroid analogue of total unimodularity. As a consequence we obtain a characterization of matroid planarity in terms of this total-unimodularity analogue.

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arXiv2021

Linking numbers, quandles and groups

We introduce a quandle invariant of classical and virtual links, denoted $Q_{tc} (L)$. This quandle has the property that $Q_{tc} (L) \cong Q_{tc} (L')$ if and only if the components of $L$ and $L'$ can be indexed in such a way that $L=K_1 \cup \dots \cup K_μ$, $L'=K'_1 \cup \dots \cup K'_μ$ and for each index $i$, there is a multiplier $ε_i \in \{-1,1\}$ that connects virtual linking numbers over $K_i$ in $L$ to virtual linking numbers over $K'_i$ in $L'$: $\ell_{j/i}(K_i,K_j)= ε_i \ell_{j/i}(K'_i,K'_j)$ for all $j \neq i$. We also extend to virtual links a classical theorem of Chen, which relates linking numbers to the nilpotent quotient $G(L)/G(L)_3$.

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arXiv2020

Multivariate Alexander quandles, II. The involutory medial quandle of a link (corrected)

Joyce showed that for a classical knot $K$, the involutory medial quandle $\text{IMQ}(K)$ is isomorphic to the core quandle of the homology group $H_1(X_2)$, where $X_2$ is the cyclic double cover of $\mathbb S ^3$, branched over $K$. It follows that $|\text{IMQ}(K)| = | \det K |$. In the present paper, the extension of Joyce's result to classical links is discussed. Among other things, we show that for a classical link $L$ of $μ\geq 2$ components, the order of the involutory medial quandle is bounded as follows: \[ \frac{μ| \det L |}{2} \geq |\text{IMQ}(L)| \geq \frac{ μ| \det L |} {2^{μ-1}}. \] In particular, $\text{IMQ}(L)$ is infinite if and only if $\det L =0$. We also show that in general, $\text{IMQ}(L)$ is a strictly stronger invariant than $H_1(X_2)$. That is, if $L$ and $L'$ are links with $\text{IMQ}(L) \cong \text{IMQ}(L')$, then $H_1(X_2) \cong H_1(X'_2)$; but it is possible to have $H_1(X_2) \cong H_1(X'_2)$ and $\text{IMQ}(L) \not \cong \text{IMQ}(L')$. In fact, it is possible to have $X_2 \cong X'_2$ and $\text{IMQ}(L) \not \cong \text{IMQ}(L')$.

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arXiv2020

Matroids that classify forests

Elementary arguments show that a tree or forest is determined (up to isomorphism) by binary matroids defined using the adjacency matrix.

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arXiv2020

Link mutations and Goeritz matrices

Extending theorems of J. E. Greene [Invent. Math. 192 (2013), 717-750] and A. S. Lipson [Enseign. Math. (2) 36 (1990), 93-114], we prove that the equivalence class of a classical link L under mutation is determined by Goeritz matrices associated to diagrams of L.

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arXiv2019

Multivariate Alexander quandles, III. Sublinks

If $L$ is a classical link then the multivariate Alexander quandle, $Q_A(L)$, is a substructure of the multivariate Alexander module, $M_A(L)$. In the first paper of this series we showed that if two links $L$ and $L'$ have $Q_A(L) \cong Q_A(L')$, then after an appropriate re-indexing of the components of $L$ and $L'$, there will be a module isomorphism $M_A(L) \cong M_A(L')$ of a particular type, which we call a"Crowell equivalence." In the present paper we show that $Q_A(L)$ (up to quandle isomorphism) is a strictly stronger link invariant than $M_A(L)$ (up to re-indexing and Crowell equivalence). This result follows from the fact that $Q_A(L)$ determines the $Q_A$ quandles of all the sublinks of $L$, up to quandle isomorphisms.

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