Sergei Chmutov, Michael Cap Khoury, Alfred Rossi
We describe the Polyak-Viro arrow diagram formulas for the coefficients of the Conway polynomial. As a consequence, we obtain the Conway polynomial as a state sum over some subsets of the crossings of the knot diagram. It turns out to be a simplification of a special case of Jaeger's state model for the HOMFLY polynomial.
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Combinatorial Goussarov-Polyak-Viro Formulas for the Linking Number and Low Degree Coefficients of the Conway Polynomial
本文利用基本线性代数，以Gauss图向量空间的基显式描述了环绕数和Conway多项式低次系数的GPV映射，提供了一种更基础的计算方法。
In this paper, it is shown that there are no nonconstant Goussarov-Polyak-Viro finite-type invariants that are invariant under the virtualization move. As an immediate corollary, we obtain the theorem which states none of the Birman coefficients of the Jones-Kauffman polynomial are of GPV finite type.
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 formulas always 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_3^1.
Goussarov, Polyak, and Viro proved that finite type invariants of knots are ``finitely multi-local'', meaning that on a knot diagram, sums of quantities, defined by local information, determine the value of the knot invariant. The result implies the existence of Gauss diagram combinatorial formulas for finite type invariants. This article presents a simplified account of the original approach. The simplifications provide an easy generalization to the cases of pure tangles and pure braids. The associated problem on group algebras is introduced and used to prove the existence of ``multi-local wo…
Polyak-Viro type formula for the Milnor triple linking number of link diagrams with multiple-crossings
本文提出一个适用于多重交叉链环图的 Polyak-Viro 型公式，用于计算 Milnor 三重链接数，其证明基于 Brooks-Komendarczyk 思想，但通过显式计算 Y-图构型空间积分，拓展了原公式的适用范围。
In the context of finite type invariants, Stanford introduced a family of equivalence relations on knots defined by the lower central series of the pure braid groups and characterized the finite type invariants in terms of the structure of the braid groups. It is known that this equivalence and Ohyama's equivalence defined by a local move are equivalent. On the other hand, in the virtual knot theory, the concept of Ohyama's equivalence was extended by Goussarov-Polyak-Viro, which called an $n$-equivalence. In this paper we extend Stanford's equivalence to virtual knots and virtual string links…
Goussarov-Polyak-Viro Conjecture for degree three case
本文显式构造长虚拟纽结三次 Vassiliev 不变量的七个 Gauss 图公式及经典纽结的 23 个公式，证明三次情形下的 Goussarov-Polyak-Viro 猜想。
Goussarov-Polyak-Viro type formulas for $(4k-1)$-dimensional knots and links in $\mathbb{R}^{6k}$
本文借鉴经典纽结有限型不变量的高斯图表示思路，推导了高维球面光滑嵌入的不变量组合公式，给出了Haefliger不变量的对应公式，并证明了相关高维辫子空间的同伦收缩核猜想。
Gauss diagram formulae for Vassiliev invariants from Kauffman polynomial
本文构建 Dubrovnik 版 Kauffman 多项式的状态模型，并据 Chmutov-Polyak 方法推导 Vassiliev 不变量的 Gauss 图公式，利用箭头图恒等式重现三阶经典公式。
We refine the Polyak-Viro Gauss diagram formula for the Vassiliev invariant of order two in a very simple way for the 2-cable of a framed long knot. Surprisingly, the resulting isotopy invariant of framed knots can detect already the non-invertibility of knots. This makes the natural generalization of our invariant for all n-component string link satellites of a framed long knot to a candidate for distinguishing all classical knots.
This paper provides the complete table of prime knot projections with their mirror images, without redundancy, up to eight double points systematically thorough a finite procedure by flypes. In this paper, we show how to tabulate the knot projections up to eight double points by listing tangles with at most four double points by an approach with respect to rational tangles of J. H. Conway. In other words, for a given prime knot projection of an alternating knot, we show how to enumerate possible projections of the alternating knot. Also to tabulate knot projections up to ambient isotopy, we in…
Using the recent Gauss diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of the polynomials of Alexander/Conway, Jones, HOMFLY, Brandt-Lickorish-Millett-Ho and Kauffman there are only finitely many positive knots with the same polynomial and no positive knot with trivial polynomial. We also discuss an extension of the Bennequin inequality, showing that t…
Using the Fiedler-Polyak-Viro Gauss diagram formulas we study the Vassiliev invariants of degree 2 and 3 on almost positive knots. As a consequence we show that the number of almost positive knots of given genus or unknotting number grows polynomially in the crossing number, and also recover and extend, inter alia to their untwisted Whitehead doubles, previous results on the polynomials and signatures of such knots. In particular, we prove that there are no achiral almost positive knots and classify all almost positive diagrams of the unknot. We give an application to contact geometry (Legendr…
From integrals to combinatorial formulas of finite type invariants -- a case study
本文将 Casson 纽结不变量的位形空间积分局部化，无需对合技巧即可证明面项消失，并由此推出正则图及花瓣图等多重交叉图的箭头图公式，给出 übercrossing 数新下界。
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