David Cimasoni
We give a geometric construction of the multivariable Conway potential function for colored links. In the case of a single color, it is Kauffman's definition of the Conway polynomial in terms of a Seifert matrix.
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In groundbreaking work from 2004, Cimasoni gave a geometric computation of the multivariable Conway potential function in terms of a generalization of a Seifert surface for a link called a C-complex. Lemma 3 of that paper provides a family of moves which relates any two C-complexes for a fixed link. This allows for an approach to studying links from the point of view of C-complexes and in following papers it has been used to derive invariants. This lemma is false. We present counterexamples, a correction with detailed proof, and an analysis of the consequences of this error on subsequent works…
We show two results about the Conway potential function which is known as the normalized multivariable Alexander polynomial. We first show that the Conway potential function introduced by Kauffman in "Formal Knot Theory" is indeed a link invariant. Next we show that Kauffman's potential function equals Hartley's potential function. We will prove it by using Murakami's axioms for the multivariable Alexander polynomial.
We give a closed formula for the multivariable Conway potential function of any graph link in a homology sphere. As corollaries, we answer three questions by Walter Neumann about graph links.
The Conway potential function (CPF) for colored links is a convenient version of the multi-variable Alexander-Conway polynomial. We give a skein characterization of CPF, much simpler than the one by Murakami. In particular, Conway's `smoothing of crossings' is not in the axioms. The proof uses a reduction scheme in a twisted group-algebra $\mathbb P_nB_n$, where $B_n$ is a braid group and $\mathbb P_n$ is a domain of multi-variable rational fractions. The proof does not use computer algebra tools. An interesting by-product is a characterization of the Alexander-Conway polynomial of knots.
We show how Conway's multivariable potential function can be constructed using braids and the reduced Gassner representation. The resulting formula is a multivariable generalization of a construction, due to Kassel-Turaev, of the Alexander-Conway polynomial in terms of the Burau representation. Apart from providing an efficient method of computing the potential function, our result also removes the sign ambiguity in the current formulas which relate the multivariable Alexander polynomial to the reduced Gassner representation. We also relate the distinct definitions of this representation which…
A weight system is defined from the (multivariable) Conway potential function. We also show that it can be calculated recursively by using five axioms.
We study relations between the Alexander-Conway polynomial $\nabla_L$ and Milnor higher linking numbers of links from the point of view of finite-type (Vassiliev) invariants. We give a formula for the first non-vanishing coefficient of $\nabla_L$ of an m-component link L all of whose Milnor numbers $μ_{i_1... i_p}$ vanish for $p\le n$. We express this coefficient as a polynomial in Milnor numbers of L. Depending on whether the parity of n is odd or even, the terms in this polynomial correspond either to spanning trees in certain graphs or to decompositions of certain 3-graphs into pairs of spa…
The Links--Gould invariant $\mathrm{LG}(L ; t_0, t_1)$ of a link $L$ is a two-variable quantum generalization of the Alexander--Conway polynomial $Δ_L(t)$ and has been shown to share some of its most geometric features in several recent works. Here we suggest that $\mathrm{LG}$ likely shares another of the Alexander polynomial's most distinctive - and mysterious - properties: for alternating links, the coefficients of the Links-Gould polynomial alternate and appear to form a log-concave two-indexed sequence with no internal zeros. The former was observed by Ishii for knots with up to 10 crossi…
Two-variable Conway polynomial and Cochran's derived invariants
本文研究链环的双变量Conway多项式与Cochran导出不变量的关系，在环绕数为0和1两种情形下给出系数$α_{1,2k-1}$的公式，并用Seifert矩阵提供新证明。
We develop an algorithm for computing generalized Seifert matrices for colored links given as closures of colored braids. The algorithm has been implemented by the second author as a computer program called Clasper. Clasper also outputs the Conway potential function, the multivariable Alexander polynomial and the Cimasoni-Florens signatures of a link, and displays a visualization of the C-complex used for producing the generalized Seifert matrices.
Satellites and invariants of links
本文肯定地解决了Arnold-Moffatt纲领中关于存在非环绕数函数的可缆化有限型链环不变量的问题，并推广至任意卫星，通过研究Conway位势函数的低次系数完成证明。
Slope and concordance of links
该研究聚焦带特异分量的有色链环同痕不变量斜率，证明其在有色协边下具有不变性，并给出基于C-复形与广义Seifert形式的斜率计算公式，丰富了链环不变量相关研究。
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 simil…
In this paper we construct a multivariable link invariant arising from the quantum group associated to the special linear Lie superalgebra sl(2|1). The usual quantum group invariant of links associated to (generic) representations of sl(2|1) is trivial. However, we modify this construction and define a nontrivial link invariant. This new invariant can be thought of as a multivariable version of the Links-Gould invariant. We also show that after a variable reduction our invariant specializes to the Conway potential function, which is a version of the multivariable Alexander polynomial.
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