Polyak Type Equations for Virtual Arrow Diagram Invariants in the Annulus

Mortier, Arnaud

doi:10.1142/s021821651350034x

Cited by 6 publications

(14 citation statements)

References 10 publications

(17 reference statements)

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“…the elements of A that define knot invariants. A result of this kind was already proved by different means in [14,15]. It is easy to see that the map d constructed in these earlier works is isomorphic with our map d Λ .…”

Section: The Stokes Formulasupporting

confidence: 63%

Finite-type 1-cocycles of knots given by Polyak–Viro Formulas

Mortier

2015

J. Knot Theory Ramifications

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We present a new method to produce simple formulas for 1-cocycles of knots over the integers, inspired by Polyak-Viro's formulas for finite-type knot invariants. We conjecture that these 1-cocycles represent finite-type cohomology classes in the sense of Vassiliev. An example of degree 3 is studied, and shown to coincide over Z 2 with the Teiblum-Turchin cocycle v 1 3 .The study of the topology of the space of knots was initiated in 1990 by Vassiliev [23], who defined finite-type cohomology classes by applying ideas from the finitedimensional affine theory of plane arrangements to the infinite-dimensional theory of long knots in 3-space. Here, of finite type means of finite complexity, in some sense, hence hopefully computable. Independently, outstanding general results on the topology of knot spaces have been obtained by Hatcher [11], leading to the idea that higher dimensional invariants of knots -in particular, 1-cocycles -should capture information about the geometry of a knot (see [7]). The zeroth level of Vassiliev's theory, known as finite-type knot invariants, has been extensively studied in the subsequent years [1,2,9,13]. However at the first level -that of 1-cocycles -only one example, in degree 3, has been proved to exist by Teiblum and Turchin, and then actually described by Vassiliev with a formula over Z 2 [22,25]. Since then, no progress has been made, probably because of the technicity of Vassiliev's construction and the apparent difficulty of turning it into a systematic method: indeed, it involves singularity theory with differential geometric 1540004-1 J. Knot Theory Ramifications 2015.24. Downloaded from www.worldscientific.com by CALIFORNIA INSTITUTE OF TECHNOLOGY on 06/27/16. For personal use only.

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Section: The Stokes Formulasupporting

confidence: 63%

Finite-type 1-cocycles of knots given by Polyak–Viro Formulas

Mortier

2015

J. Knot Theory Ramifications

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“…The farness constraint could be lightened, by allowing R-III close diagrams. In the case i = 0, this is harmless (there are no additional equations) thanks to [14,Lemma 3.2], and it yields all GPV invariants [14,Theorem 3.6]. For higher values of i, it would require to compute the proper ε signs to associate with Type 3-3 degeneracies, and to consider subgerms whose graphs are not necessarily trees.…”

Section: Cohomology Of Tree Diagrams and Of The Space Of Knotsmentioning

confidence: 99%

Combinatorial cohomology of the space of long knots

Mortier

2015

Algebr. Geom. Topol.

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The motivation of this work is to define cohomology classes in the space of knots that are both easy to find and to evaluate, by reducing the problem to simple linear algebra. We achieve this goal by defining a combinatorial graded cochain complex, such that the elements of an explicit submodule in the cohomology define algebraic intersections with some "geometrically simple" strata in the space of knots. Such strata are endowed with explicit co-orientations, that are canonical in some sense. The combinatorial tools involved are natural generalisations (degeneracies) of usual methods using arrow diagrams.

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“…The only difficult part of the statement is the invariance under RIII and it is due to A. Mortier. It can be found in [22] where it is stated as an equivalence in the context of virtual knots. However, the arguments adapt verbatim to the case of virtual string links.…”

Section: Gauss Diagram Formulae For Virtual and Welded String Linksmentioning

confidence: 99%

“…Now, such a map does not, in general, factor through the generalized Reidemeister moves. We recall below a simple criterion, due to Mortier [22], which gives a sufficient condition for getting a virtual string link invariant in this way. To state this criterion, we need a few more definitions.…”

Section: Gauss Diagram Formulae For Virtual and Welded String Linksmentioning

confidence: 99%

On usual, virtual and welded knotted objects up to homotopy

Audoux¹,

Bellingeri²,

Meilhan³

et al. 2017

J. Math. Soc. Japan

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We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or selfvirtualizations. We provide a number of results which point out the differences between these various notions. The proofs are mainly based on the techniques of Gauss diagram formulae.• P n and SL n stand for the (usual) sets of pure braids and string links on n strands, and the prefix v and w refer to their virtual and welded counterpart, respectively; • the superscripts sv and sc refer respectively to the equivalence relations generated by self-virtualization and self-crossing change.

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Polyak Type Equations for Virtual Arrow Diagram Invariants in the Annulus

Cited by 6 publications

References 10 publications

Finite-type 1-cocycles of knots given by Polyak–Viro Formulas

Finite-type 1-cocycles of knots given by Polyak–Viro Formulas

Combinatorial cohomology of the space of long knots

On usual, virtual and welded knotted objects up to homotopy

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