Alexander Representation of Tangles

Bigelow, Stephen; Cattabriga, Alessia; Florens, Vincent

doi:10.1007/s40306-015-0134-z

Cited by 5 publications

(7 citation statements)

References 14 publications

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“…If these diagrams do not have any virtual crossing, the construction holds for usual links, and we obtain a purely local description of the usual Alexander polynomial. This extends the construction of the Alexander representation by the second author, Bigelow and Cattabriga, see [BCF15].…”

Section: Introductionsupporting

confidence: 69%

Alexander invariants of ribbon tangles and planar algebras

Damiani¹,

Florens²

2018

J. Math. Soc. Japan

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Ribbon tangles are proper embeddings of tori and cylinders in the 4ball B 4 , "bounding" 3-manifolds with only ribbon disks as singularities. We construct an Alexander invariant 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 RibG 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 CobG over the operad of smooth cobordisms, inspired by diagrammatic planar algebras introduced by Jones [Jon99], and prove that the invariant 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 A and of the algebra CobG, and observe that our construction is a topological incarnation of the Alexander invariant of Archibald [Arc10]. When restricted to diagrams without virtual crossings, A provides a purely local description of the usual Alexander poynomial of links, and extends the construction by Bigelow, Cattabriga and the second author [BCF15].

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Section: Introductionsupporting

confidence: 69%

Alexander invariants of ribbon tangles and planar algebras

Damiani¹,

Florens²

2018

J. Math. Soc. Japan

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“…Our constructions are also related to the work of Bigelow, Cattabriga and the first author [BCF12], which provides a functorial extension of the Alexander polynomial to the category of tangles instead of the category of cobordisms. To describe this relation, let TangCob be the monoidal category whose objects are pairs of non-negative integers (g, n) -corresponding to surfaces F g with n punctures -and whose morphisms are cobordisms with tangles inside.…”

Section: Introductionmentioning

confidence: 99%

“…When G is the infinite cyclic group generated by t, the usual category Tang + of oriented tangles in the standard ball can be regarded as a subcategory of TangCob G by only considering those representations of tangle exteriors that send any oriented meridian to the generator t. The functors A and R constructed in this paper could be extended to the category TangCob G using similar methods, but with more technicality. When G is infinite cyclic, the restriction of the resulting functor A : TangCob G → grMod Z[G],±G to Tang + would coincide with the "Alexander representation of tangles" constructed in [BCF12]. We also mention Archibald's extension of the Alexander polynomial [Arc10], which is based on diagrammatic presentations of tangles: her invariant seems to be very close to the invariant constructed in [BCF12] and it is stronger since it is defined without ambiguity in ±G.…”

Section: Introductionmentioning

confidence: 99%

“…When G is infinite cyclic, the restriction of the resulting functor A : TangCob G → grMod Z[G],±G to Tang + would coincide with the "Alexander representation of tangles" constructed in [BCF12]. We also mention Archibald's extension of the Alexander polynomial [Arc10], which is based on diagrammatic presentations of tangles: her invariant seems to be very close to the invariant constructed in [BCF12] and it is stronger since it is defined without ambiguity in ±G.…”

Section: Introductionmentioning

confidence: 99%

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

Florens¹,

Massuyeau²

2016

Enseign. Math.

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Abstract. Let F be a field and let G ⊂ F \ {0} be a multiplicative subgroup. We consider the category CobG of 3-dimensional cobordisms equipped with a representation of their fundamental group in G, and the category Vect F,±G of F-linear maps defined up to multiplication by an element of ±G. Using the elementary theory of Reidemeister torsions, we construct a "Reidemeister functor" from CobG to Vect F,±G . In particular, when the group G is free abelian and F is the field of fractions of the group ring 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 ⊂ F \ {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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“…Florens[BCF12] and Damiani-Florens[DV16] work with suitable generalisations of Alexander matrices. ∇ s T from this paper fits into this collection of invariants, as it is based on the purely combinatorial definition of the classical Alexander polynomial via Kauffman states and Alexander codes.…”

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confidence: 99%

Kauffman states and Heegaard diagrams for tangles

Zibrowius

2019

Algebr. Geom. Topol.

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We define polynomial tangle invariants ∇ s T via Kauffman states and Alexander codes and investigate some of their properties. In particular, we prove symmetry relations for ∇ s T of 4-ended tangles and deduce that the multivariable Alexander polynomial is invariant under Conway mutation. The invariants ∇ s T can be interpreted naturally via Heegaard diagrams for tangles. This leads to a categorified version of ∇ s T : a Heegaard Floer homology HFT for tangles, which we define as a bordered sutured invariant. We discuss a bigrading on HFT and prove symmetry relations for HFT of 4-ended tangles that echo those for ∇ s T .Kauffman states and Heegaard diagrams for tangles 3 of L and we call R the mutating tangle in this mutation. If L is oriented, we choose an orientation of L that agrees with the one for L outside of R. If this means that we need to reverse the orientation of the two open components of R, then we also reverse the orientation of all other components of R during the mutation; otherwise we do not change any orientation. For an alternative, but equivalent definition, see definition 3.3 and remark 3.4.Theorem 0.2 along with the glueing formula for ∇ s T gives rise to the following result. Corollary 0.4 (3.5) The multivariate Alexander polynomial is invariant under Conway mutation after identifying the variables corresponding to the two open strands of the mutating tangle.This result has long been known for the univariate Alexander polynomial, see for example [LM87, proposition 11], but I have been unable to find a corresponding result for the multivariate polynomial in the literature. The fact that mutation invariance follows so easily from the symmetry relations of theorem 0.2 suggests that ∇ T is well-suited for studying the "local behaviour" of the Alexander polynomial. The homological tangle invariant HFTHeegaard Floer homology theories were first defined by Ozsváth and Szabó in 2001 [OS01]. With an oriented, closed 3-dimensional manifold M , they associated a family of homological invariants, the simplest of which is denoted by HF(M). Given an oriented (null-homologous) knot or link L in M , Ozsváth and Szabó, and independently J. Rasmussen, then defined filtrations on the chain complexes which give rise to the respective flavours of knot and link Floer homology [OS03a, Ras03, OS05], the simplest of which is denoted by HFL(L). The Alexander polynomial can be recovered from these groups as the graded Euler characteristic.Given corollary 0.4, it is only natural to ask for a Heegaard-Floer theoretic categorification of ∇ s T . To this end, we define a homology theory HFT as follows: given a Heegaard diagram for a tangle T (see definition 4.1) along with a site s of T , we define a finitely generated Abelian group which comes with two gradings: a relative homological Z-grading and an Alexander grading, which is an additional relative Z-grading for each component of the tangle:CFT(T, s) = h∈Z ←homological grading a∈Z |T| ←Alexander grading CFT h (T, s, a).

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Alexander Representation of Tangles

Cited by 5 publications

References 14 publications

Alexander invariants of ribbon tangles and planar algebras

Alexander invariants of ribbon tangles and planar algebras

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

Kauffman states and Heegaard diagrams for tangles

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