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Lorenzo Traldi

Publications and source records attributed to Lorenzo Traldi.

At least 37 records · Page 2Linked to original sources
arXiv2019

A Characterization of Circle Graphs in Terms of Multimatroid Representations

The isotropic matroid $M[IAS(G)]$ of a looped simple graph $G$ is a binary matroid equivalent to the isotropic system of $G$. In general, $M[IAS(G)]$ is not regular, so it cannot be represented over fields of characteristic $\neq 2$. The ground set of $M[IAS(G)]$ is denoted $W(G)$; it is partitioned into 3-element subsets corresponding to the vertices of $G$. When the rank function of $M[IAS(G)]$ is restricted to subtransversals of this partition, the resulting structure is a multimatroid denoted $\mathcal{Z}_{3}(G)$. In this paper we prove that $G$ is a circle graph if and only if for every field $\mathbb{F}$, there is an $\mathbb{F}$-representable matroid with ground set $W(G)$, which defines $\mathcal{Z}_{3}(G)$ by restriction. We connect this characterization with several other circle graph characterizations that have appeared in the literature.

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arXiv2018

A note on Dehn colorings and invariant factors

If $A$ is an abelian group and $ϕ$ is an integer, let $A(ϕ)$ be the subgroup of $A$ consisting of elements $a \in A$ such that $ϕ\cdot a=0$. We prove that if $D$ is a diagram of a classical link $L$ and $0=ϕ_0,ϕ_1,\dots,ϕ_{n-1}$ are the invariant factors of an adjusted Goeritz matrix of $D$, then the group $\mathcal{D}_{A}(D)$ of Dehn colorings of $D$ with values in $A$ is isomorphic to the direct product of $A$ and $A=A(ϕ_{0}),A(ϕ_1),\dots,A(ϕ_{n-1})$. It follows that the Dehn coloring groups of $L$ are isomorphic to those of a connected sum of torus links $T_{(2,ϕ_1)} \text{ }\# \text{ } \cdots \text{ } \# \text{ } T_{(2,ϕ_{n-1})}$.

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arXiv2018

Multivariate Alexander colorings

We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module $M$ over the Laurent polynomial ring $Λ_μ=\mathbb{Z}[t_1^{\pm1},\dots,t_μ^{\pm1}]$. If $D$ is a diagram of a link $L$ with $μ$ components, then the colorings of $D$ with values in $M$ form a $Λ_μ$-module $\mathrm{Color}_A(D,M)$. Extending a result of Inoue [Kodai Math.\ J.\ 33 (2010), 116-122], we show that $\mathrm{Color}_A(D,M)$ is isomorphic to the module of $Λ_μ$-linear maps from the Alexander module of $L$ to $M$. In particular, suppose $M$ is a field and $φ:Λ_μ \to M$ is a homomorphism of rings with unity. Then $φ$ defines a $Λ_μ$-module structure on $M$, which we denote $M_φ$. We show that the dimension of $\mathrm{Color}_A(D,M_φ)$ as a vector space over $M$ is determined by the images under $φ$ of the elementary ideals of $L$. This result applies in the special case of Fox tricolorings, which correspond to $M=GF(3)$ and $φ(t_i) \equiv-1$. Examples show that even in this special case, the higher Alexander polynomials do not suffice to determine $|\mathrm{Color}_A(D,M_φ)|$; this observation corrects erroneous statements of Inoue [J. Knot Theory Ramifications 10 (2001), 813-821; op. cit.].

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arXiv2018

Goeritz and Seifert Matrices from Dehn Presentations

The Goeritz matrix of a link is obtained from the Jacobian matrix of a modified Dehn presentation associated to a diagram using Fox's free differential calculus. When the diagram is special the Seifert matrix can also be determined from the presentation.

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arXiv2017

Fourientation activities and the Tutte polynomial

A fourientation of a graph $G$ is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. We may naturally view fourientations as a mixture of subgraphs and graph orientations where unoriented and bioriented edges play the role of absent and present subgraph edges, respectively. Building on work of Backman and Hopkins (2015), we show that given a linear order and a reference orientation of the edge set, one can define activities for fourientations of $G$ which allow for a new 12 variable expansion of the Tutte polynomial $T_G$. Our formula specializes to both the Las Vergnas (1984) orientation activites expansion of the Tutte polynomial and the generalized activities expansion of Gordon and Traldi (1990).

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arXiv2017

Isotropic matroids III: Connectivity

The isotropic matroid $M[IAS(G)]$ of a graph $G$ is a binary matroid, which is equivalent to the isotropic system introduced by Bouchet. In this paper we discuss four notions of connectivity related to isotropic matroids and isotropic systems. We show that the isotropic system connectivity defined by Bouchet is equivalent to vertical connectivity of $M[IAS(G)]$, and if $G$ has at least four vertices, then $M[IAS(G)]$ is vertically 5-connected if and only if $G$ is prime (in the sense of Cunningham's split decomposition). We also show that $M[IAS(G)]$ is $3$-connected if and only if $G$ is connected and has neither a pendant vertex nor a pair of twin vertices. Our most interesting theorem is that if $G$ has $n\geq7$ vertices then $M[IAS(G)]$ is not vertically $n$-connected. This abstract-seeming result is equivalent to the more concrete assertion that $G$ is locally equivalent to a graph with a vertex of degree $<\frac{n-1}{2}$.

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arXiv2016

Isotropic matroids I: Multimatroids and neighborhoods

Several properties of the isotropic matroid of a looped simple graph are presented. Results include a characterization of the multimatroids that are associated with isotropic matroids and several ways in which the isotropic matroid of G incorporates information about graphs locally equivalent to G. Specific results of the latter type include a characterization of graphs that are locally equivalent to bipartite graphs, a direct proof that two forests are isomorphic if and only if their isotropic matroids are isomorphic, and a way to express local equivalence indirectly, using only edge pivots.

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arXiv2016

Notes on a theorem of Naji

We present a new proof of an algebraic characterization of circle graphs due to W. Naji. For bipartite graphs, Naji's theorem is equivalent to an algebraic characterization of planar matroids due to J. Geelen and B. Gerards. Naji's theorem also yields an algebraic characterization of permutation graphs.

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arXiv2015

Splitting cubic circle graphs

We show that every 3-regular circle graph has at least two pairs of twin vertices; consequently no such graph is prime with respect to the split decomposition. We also deduce that up to isomorphism, K_4 and K_{3,3} are the only 3-connected, 3-regular circle graphs.

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arXiv2015

The transition matroid of a 4-regular graph: an introduction

Given a 4-regular graph $F$, we introduce a binary matroid $M_τ(F)$ on the set of transitions of $F$. Parametrized versions of the Tutte polynomial of $M_τ(F)$ yield several well-known graph and knot polynomials, including the Martin polynomial, the homflypt polynomial, the Kauffman polynomial and the Bollobás-Riordan polynomial.

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arXiv2014

Binary matroids and local complementation

We introduce a binary matroid M(IAS(G)) associated with a looped simple graph G. M(IAS(G)) classifies G up to local equivalence, and determines the delta-matroid and isotropic system associated with G. Moreover, a parametrized form of its Tutte polynomial yields the interlace polynomials of G.

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arXiv2013

Interlace polynomials and Tutte polynomials

Let G be a graph with adjacency matrix A(G). Consider the matrix IA(G)=(I | A(G)), where I is the identity matrix, and let M(IA(G)) be the binary matroid represented by IA(G). Then suitably parametrized versions of the Tutte polynomial of M(IA(G)) yield the interlace polynomials of G, introduced by Arratia, Bollobás and Sorkin [J. Combin. Theory Ser. B 92 (2004) 199-233; Combinatorica 24 (2004) 567-584]. Interlace polynomials subsequently introduced by other authors may be obtained from parametrized Tutte polynomials of the binary matroid represented by (I | A(G) | I+A(G)).

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arXiv2012

The adjacency matroid of a graph

If $G$ is a looped graph, then its adjacency matrix represents a binary matroid $M_{A}(G)$ on $V(G)$. $M_{A}(G)$ may be obtained from the delta-matroid represented by the adjacency matrix of $G$, but $M_{A}(G)$ is less sensitive to the structure of $G$. Jaeger proved that every binary matroid is $M_{A}(G)$ for some $G$ [Ann. Discrete Math. 17 (1983), 371-376]. The relationship between the matroidal structure of $M_{A}(G)$ and the graphical structure of $G$ has many interesting features. For instance, the matroid minors $M_{A}(G)-v$ and $M_{A}(G)/v$ are both of the form $M_{A}(G^{\prime}-v)$ where $G^{\prime}$ may be obtained from $G$ using local complementation. In addition, matroidal considerations lead to a principal vertex tripartition, distinct from the principal edge tripartition of Rosenstiehl and Read [Ann. Discrete Math. 3 (1978), 195-226]. Several of these results are given two very different proofs, the first involving linear algebra and the second involving set systems or delta-matroids. Also, the Tutte polynomials of the adjacency matroids of $G$ and its full subgraphs are closely connected to the interlace polynomial of Arratia, Bollobás and Sorkin [Combinatorica 24 (2004), 567-584].

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arXiv2012

On the interlace polynomials

The generating function that records the sizes of directed circuit partitions of a connected 2-in, 2-out digraph D can be determined from the interlacement graph of D with respect to a directed Euler circuit; the same is true of the generating functions for other kinds of circuit partitions. The interlace polynomials of Arratia, Bollobás and Sorkin [J. Combin. Theory Ser. B 92 (2004) 199-233; Combinatorica 24 (2004) 567-584] extend the corresponding functions from interlacement graphs to arbitrary graphs. We introduce a multivariate interlace polynomial that is an analogous extension of a multivariate generating function for undirected circuit partitions of undirected 4-regular graphs. The multivariate polynomial incorporates several different interlace polynomials that have been studied by different authors, and its properties include invariance under a refined version of local complementation and a simple recursive definition.

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arXiv2012

Interlacement in 4-regular graphs: a new approach using nonsymmetric matrices

Let F be a 4-regular graph with an Euler system C. We introduce a simple way to modify the interlacement matrix of C so that every circuit partition P of F has an associated modified interlacement matrix M(C,P). If C and C' are Euler systems of F then M(C,C') and M(C',C) are inverses, and for any circuit partition P, M(C',P)=M(C',C)M(C,P). This machinery allows for short proofs of several results regarding the linear algebra of interlacement.

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