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

Abstract

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.

Combinatorial cohomology of the space of long knots

“…This elementary proof should be compared with that of [15,Lemma 1.9]. There, the result was deeply related with the fact that the germ was topological.…”

“…We construct two unbased diagrams by replacing the arrow (a, b) with (a, ∞) (resp. (b, c) with (∞, c)) while forgetting the point ∞, and give them signs depending only on the relative position of a, b and c in the cyclic order -see the rule on Fig.7 This theorem can be proved by analysing the presentation of rot from [7, Fig.144], and that of FH given by Fox [9] from the viewpoint of Gauss diagrams -as in the proof of [15,Theorem 3.3]. Reidemeister farness is crucial in the proof, not only for the theory to work properly, but to have a good control of the non-trivial contributions to the integrals.…”

“…Reidemeister farness is crucial in the proof, not only for the theory to work properly, but to have a good control of the non-trivial contributions to the integrals. For example, the 1-cocycle formula from [15,Theorem 3.2], which allows R-III close diagrams, is impossible [to us] to integrate directly on the Fox-Hatcher loop, because of uncontrollable contributions.…”

“…For i = 0, the image of this map consists of invariants induced by homogeneous Goussarov-Polyak-Viro formulas for long virtual knots [10]. For i = 1, the image consists of arrow-germ formulas as defined in [15]. Remark 3.12.…”

A Kontsevich integral of order 1

“…Since Vassiliev [21] and Hatcher [12] (followed by Budney [4] and Budney-Cohen [5]) introduced an interest for the topology of the space of knots, it is remarkable how few attempts have been made to build actual realisations of 1-cocycles-the simplest topological invariants after classical "knot invariants", and at the same time how different these attempts have been from each other: see [22], [20,19], [15,16], [9,10,11]. This is not the first attempt to build 1-cocycles by means of integrals: Sakai's method in [20,19] uses configuration space integrals-see also [6,3].…”