arXiv subjects

Vincent Florens

Publications and source records attributed to Vincent Florens.

15 recordsLinked to original sources

arXiv2022

Slope and concordance of links

The slope is an isotopy invariant of colored links with a distinguished component, initially introduced by the authors to describe an extra correction term in the computation of the signature of the splice. It appeared to be closely related to several classical invariants, such as the Conway potential function or the Kojima eta-function (defined for two-components links). In this paper, we prove that the slope is invariant under colored concordance of links. Besides, we present a formula to compute the slope in terms of C-complexes and generalized Seifert forms.

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arXiv2021

A Slope invariant and the A-polynomial of knots

The A-polynomial is a knot invariant related to the space of $SL_2(\mathbb{C})$ representations of the knot group. In this paper our interests lies in the logarithmic Gauss map of the A-polynomial. We develop a homological point of view on this slope by extending the constructions of Degtyarev, the second author and Lecuona to the setting of non-abelian representations. It defines a rational function on the character variety, which unifies various known invariants such as the change of curves in the Reidemeister function, the modulus of boundary-parabolic representations, the boundary slope of some incompressible surfaces embedded in the exterior of the knot $K$ or equivalently the slopes of the sides of the Newton polygon of the A-polynomial $A_K$. We also present a method to compute $s_K$ in terms of Alexander matrices and Fox calculus.

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arXiv2021

Torsions and intersection forms of 4-manifolds from trisection diagrams

Gay and Kirby introduced trisections which describe any closed oriented smooth 4-manifold $X$ as a union of three four-dimensional handlebodies. A trisection is encoded in a diagram, namely three collections of curves in a closed oriented surface $Σ$, guiding the gluing of the handlebodies. Any morphism $φ$ from $π_1(X)$ to a finitely generated free abelian group induces a morphism on $π_1(Σ)$. We express the twisted homology and Reidemeister torsion of $(X;φ)$ in terms of the first homology of $(Σ;φ)$ and the three subspaces generated by the collections of curves. We also express the intersection form of $(X;φ)$ in terms of the intersection form of $(Σ;φ)$.

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arXiv2020

Slopes of links and signature formulas

We present a new invariant, called slope, of a colored link in an integral homology sphere and use this invariant to complete the signature formula for the splice of two links. We develop a number of ways of computing the slope and a few vanishing results. Besides, we discuss the concordance invariance of the slope and establish its close relation to the Conway polynomials, on the one hand, and to the Kojima--Yamasaki $η$-function (in the univariate case) and Cochran invariants, on the other hand.

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arXiv2018

Slopes and signatures of links

We define the slope of a colored link in an integral homology sphere, associated to admissible characters on the link group. Away from a certain singular locus, the slope is a rational function which can be regarded as a multivariate generalization of the Kojima--Yamasaki $η$-function. It is the ratio of two Conway potentials, provided that the latter makes sense; otherwise, it is a new invariant. The slope is responsible for an extra correction term in the signature formula for the splice of two links, in the previously open exceptional case where both characters are admissible. Using a similar construction for a special class of tangles, we formulate generalized skein relations for the signature.

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arXiv2017

A functorial extension of the Magnus representation to the category of three-dimensional cobordisms

Let $R$ be an integral domain and $G$ be a subgroup of its group of units. We consider the category $\mathbf{\mathsf{Cob}}_G$ of 3-dimensional cobordisms between oriented surfaces with connected boundary, equipped with a representation of their fundamental group in $G$. Under some mild conditions on $R$, we construct a monoidal functor from $\mathbf{\mathsf{Cob}}_G$ to the category $\mathbf{\mathsf{pLagr}}_R$ consisting of "pointed Lagrangian relations" between skew-Hermitian $R$-modules. We call it the "Magnus functor" since it contains the Magnus representation of mapping class groups as a special case. Our construction is inspired from the work of Cimasoni and Turaev on the extension of the Burau representation of braid groups to the category of tangles. It can also be regarded as a $G$-equivariant version of a TQFT-like functor that has been described by Donaldson. The study and computation of the Magnus functor is carried out using classical techniques of low-dimensional topology. When $G$ is a free abelian group and $R = Z[G]$ is the group ring of $G$, we relate the Magnus functor to the "Alexander functor" (which has been introduced in a prior work using Alexander-type invariants), and we deduce a factorization formula for the latter.

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arXiv2016

The signature of a splice

We study the behavior of the signature of colored links [Flo05, CF08] under the splice operation. We extend the construction to colored links in integral homology spheres and show that the signature is almost additive, with a correction term independent of the links. We interpret this correction term as the signature of a generalized Hopf link and give a simple closed formula to compute it.

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arXiv2016

Alexander invariants of ribbon tangles and planar algebras

Ribbon tangles are proper embeddings of tori and cylinders in the $4$-ball~$B^4$, "bounding" $3$-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathsf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. This invariant induces a functor in a certain category $\mathsf{R}ib_G$ of tangles, which restricts to the exterior powers of Burau-Gassner representation for ribbon braids, that are analogous to usual braids in this context. We define a circuit algebra $\mathsf{C}ob_G$ over the operad of smooth cobordisms, inspired by diagrammatic planar algebras introduced by Jones, and prove that the invariant $\mathsf{A}$ commutes with the compositions in this algebra. On the other hand, ribbon tangles admit diagrammatic representations, throught welded diagrams. We give a simple combinatorial description of $\mathsf{A}$ and of the algebra $\mathsf{C}ob_G$, and observe that our construction is a topological incarnation of the Alexander invariant of Archibald. When restricted to diagrams without virtual crossings, $\mathsf{A}$ provides a purely local description of the usual Alexander poynomial of links, and extends the construction by Bigelow, Cattabriga and the second author.

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arXiv2015

On complex line arrangements and their boundary manifolds

Let A be a line arrangement in the complex projective plane CP2. We define and describe the inclusion map of the boundary manifold --the boundary of a close regular neighborhood of A-- in the exterior of the arrangement. We obtain two explicit descriptions of the map induced on the fundamental groups. These computations provide a new minimal presentation of the fundamental group of the complement.

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arXiv2014

A functorial extension of the abelian Reidemeister torsions of three-manifolds

Let $\mathbb{F}$ be a field and let $G\subset \mathbb{F}\setminus \{0\}$ be a multiplicative subgroup. We consider the category $\mathcal{Cob}_G$ of $3$-dimensional cobordisms equipped with a representation of their fundamental group in $G$, and the category ${Vect}_{\mathbb{F},\pm G}$ of $\mathbb{F}$-linear maps defined up to multiplication by an element of $\pm G$. Using the elementary theory of Reidemeister torsions, we construct a "Reidemeister functor" from $Cob_G$ to ${Vect}_{\mathbb{F},\pm G}$. In particular, when the group $G$ is free abelian and $\mathbb{F}$ is the field of fractions of the group ring $\mathbb{Z}[G]$, we obtain a functorial formulation of an Alexander-type invariant introduced by Lescop for $3$-manifolds with boundary; when $G$ is trivial, the Reidemeister functor specializes to the TQFT developed by Frohman and Nicas to enclose the Alexander polynomial of knots. The study of the Reidemeister functor is carried out for any multiplicative subgroup $G\subset \mathbb{F}\setminus \{0\}$. We obtain a duality result and we show that the resulting projective representation of the monoid of homology cobordisms is equivalent to the Magnus representation combined with the relative Reidemeister torsion.

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arXiv2014

Alexander representation of tangles

A tangle is an oriented 1-submanifold of the cylinder whose endpoints lie on the two disks in the boundary of the cylinder. Using an algebraic tool developed by Lescop, we extend the Burau representation of braids to a functor from the category of oriented tangles to the category of Z[t,t^{-1}]-modules. For (1,1)-tangles (i.e., tangles with one endpoint on each disk) this invariant coincides with the Alexander polynomial of the link obtained by taking the closure of the tangle. We use the notion of plat position of a tangle to give a constructive proof of invariance in this case.

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arXiv2014

A topological invariant of line arrangements

We define a new topological invariant of line arrangements in the complex projective plane. This invariant is a root of unity defined under some combinatorial restrictions for arrangements endowed with some special torsion character on the fundamental group of their complements. It is derived from the peripheral structure on the group induced by the inclusion map of the boundary of a tubular neigborhood in the exterior of the arrangement. By similarity with knot theory, it can be viewed as an analogue of linking numbers. This is an orientation-preserving invariant for ordered arrangements. We give an explicit method to compute the invariant from the equations of the arrangement, by using wiring diagrams introduced by Arvola, that encode the braid monodromy. Moreover, this invariant is a crucial ingredient to compute the depth of a character satisfying some resonant conditions, and complete the existent methods by Libgober and the first author. Finally, we compute the invariant for extended MacLane arrangements with an additional line and observe that it takes different values for the deformation classes.

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arXiv2005

Generalized Seifert surfaces and signatures of colored links

In this paper, we use `generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in S^3. This yields an integral valued function on the m-dimensional torus, where m is the number of colors of the link. The case m=1 corresponds to the Levine-Tristram signature. We show that many remarkable properties of the latter invariant extend to this m-variable generalization: it vanishes for achiral colored links, it is `piecewise continuous', and the places of the jumps are determined by the Alexander invariants of the colored link. Using a 4-dimensional interpretation and the Atiyah-Singer G-signature theorem, we also prove that this signature is invariant by colored concordance, and that it provides a lower bound for the `slice genus' of the colored link.

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arXiv2005

Twisted Alexander polynomials of Plane Algebraic Curves

We consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is given by the determinant of its Blanchfield intersection form. Specializing in the classical case, this gives a geometrical interpretation of Libgober's divisibility Theorem. We calculate twisted polynomials for some algebraic curves and show how they can detect Zariski pairs of equivalent Alexander polynomials and that they are sensitive to nodal degenerations.

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arXiv2003

On the slice genus of links

We define Casson-Gordon sigma-invariants for links and give a lower bound of the slice genus of a link in terms of these invariants. We study as an example a family of two component links of genus h and show that their slice genus is h, whereas the Murasugi-Tristram inequality does not obstruct this link from bounding an annulus in the 4-ball.

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