Jones多项式的零点

利用数论理论证明了纽结的Jones多项式仅有可能的有理根是O,而链环的Jones多项式仅有可能的有理根是0和-1.给出了作为Jones多项式根的所有可能单位根,以及所有可能的具有平凡Mahler测度的Jones多项式.最后指出了交叉数不超过11的纽结中,只有4_1,8_9,9_(42),K11n19的Jones多项式具有平凡的Mahler测度,从而回答了林晓松提出的关于Mahler测度的一个问题.     

Strong band sum and dichromatic invariants

Strong band sum is a natural construction from links to dichromatic links. We compute Hoste and Kidwell's dichromatic link invariant of a strong band sum in terms of monochromatic invariants of the data (original link, band). It turns out that the two-variable Conway polynomial of a strong fusion only depends on the monochromatic Conway polynomial of the original link. In particular, it does not depend on the band. Cochran's series of concordance invariants is discussed in this framework. partially supported by NATO via DAAD     

Realizing Alexander polynomials by hyperbolic links

We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link having any number of components, and by infinitely many such links having at least 4 components. As a consequence, a Mahler measure minimizing polynomial, if it exists, is realized as the Alexander polynomial of a fibered hyperbolic link of at least 2 components. For a given polynomial, we also give an upper bound for the minimal hyperbolic volume of knots/links realizing the polynomial and, in the opposite direction, construct knots of arbitrarily large volume, which are arborescent, or have given free genus at least 2.     

On alternation numbers of links

We construct infinitely many hyperbolic links with x-distance far from the set of (possibly, splittable) alternating links in the concordance class of every link. A sensitive result is given for the concordance class of every (possibly, split) alternating link. Our proof uses an estimate of the τ-distance by an Alexander invariant and the topological imitation theory, both established earlier by the author.     

Conway Polynomial and Oriented Rotant Links

We prove that the Conway polynomial of oriented links is not changed by the rotation operation of Anstee, Przytycki and Rolfsen.Mathematics Subject Classifications (2000). 57M25, 57M27Supported by KBN grant No 2P03A00218     

一类纽结的Conway多项式不变量

本文研究了一类特殊纽结，证明了存在无限多个仅具有两个负交叉点的纽结，而其Ｃｏｎｗａｙ多项式非正．事实上，也给出了此类纽结的Ｃｏｎｗａｙ多项式一个一般公式．     

超额市场需求函数

本文通过研究超额需求函数与多项式的关系得：（１）对于（ｎ—１）个关于ｐ１,ｐ２…,ｐｎ－１的多项式Ｅ。（ｐ１,ｐ２,…,ｐｎ－１）,ｓ＝１,２,…,ｎ—１,若满足条件：则Ｅｓ,ｓ＝１,２,…,ｎ—１均为一个经济的超额需求函数．（２）对无穷级数ｇ（ｐ）,也存在含有两个消费者与无穷种商品的经济,使得其超额需求函数恰好为ｇ（ｐ）．     

Link polynomials derived from magnetic graphs

We introduce a graph diagram which can be regarded as a generalized link diagram. By using it, we construct two polynomial invariants for knots and links which are distinct from both the HOMFLY and the Kauffman polynomials. We show some features of the polynomials including relationships with the HOMFLY and the Kauffman polynomials.     

Mutations of links in genus 2 handlebodies

A short proof is given to show that a link in the 3-sphere and any link related to it by genus 2 mutation have the same Alexander polynomial. This verifies a deduction from the solution to the Melvin-Morton conjecture. The proof here extends to show that the link signatures are likewise the same and that these results extend to links in a homology 3-sphere.

     

Homotopy, Δ-equivalence and concordance for knots in the complement of a trivial link

Link-homotopy and self Δ-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Δ-equivalent) to a trivial link. We study link-homotopy and self Δ-equivalence on a certain component of a link with fixing the other components, in other words, homotopy and Δ-equivalence of knots in the complement of a certain link. We show that Milnor invariants determine whether a knot in the complement of a trivial link is null-homotopic, and give a sufficient condition for such a knot to be Δ-equivalent to the trivial knot. We also give a sufficient condition for knots in the complements of the trivial knot to be equivalent up to Δ-equivalence and concordance.     

Alternating link and its generalization

The alternating links give a classical class of links. They play an important role in Knot Theory. Ozsváth and Szabó introduced a quasi-alternating link which is a generalization of an alternating link. In this paper we review some results of alternating links and quasi-alternating links on the Jones polynomial and the Khovanov homology. Moreover, we introduce a long pass link. Several problems worthy of further study are provided.     

Milnor numbers, spanning trees, and the Alexander–Conway polynomial

We study relations between the Alexander–Conway polynomial _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 _L of an m-component link L all of whose Milnor numbers μ_i1…ip vanish for pn. 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 spanning trees. Our results complement determinantal formulas of Traldi and Levine obtained by geometric methods.     

Bands,tangles and linear skein theory

The process of attaching bands to links (fusion/fission) is discussed in the framework of tangle theory and linear skein theory. Formulas for skein polynomials are deduced and nontriviality results for band constructions are proved. In particular we discuss the effect of band changes like twisting. We prove that for each link and choice of two attaching arcs there are infinitely many different fusion/fission links with bands attached to these arcs. partially supported by NATO via DAAD     

Characterization of 3-bridge links with infinitely many 3-bridge spheres

In an earlier paper, the author constructed an infinite family of 3-bridge links each of which admits infinitely many 3-bridge spheres up to isotopy. In this paper, we prove that if a prime, unsplittable link L in S³ admits infinitely many 3-bridge spheres up to isotopy then L belongs to the family.     

Playing disjunctive sums is polynomial space complete

A position in a disjunctive sum of games is simply a collection of positions, one from each game: to move in a sum is to move in any one of its constituents. Sums have been studied extensively by Conway and others, and play an important rÔle in Go. It is shown that the problem of best play in a sum of trivial games is polynomial space complete. Hence it may be conjectured that there is no feasible algorithm for deriving a strategy of play in a sum from knowledge about its constituent games.     

On Conway’s potential function for colored links

The Conway potential function(CPF) for colored links is a convenient version of the multivariable 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 P_nB_n, where B_n is a braid group and 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.     

Arithmetics of 2-friezes

We consider the variant of Coxeter–Conway frieze patterns called 2-frieze. We prove that there exist infinitely many closed integral 2-friezes (i.e. containing only positive integers) provided the width of the array is bigger than 4. We introduce operations on the integral 2-friezes generating bigger or smaller closed integral 2-friezes.     

Zeros Distribution of the Jones Polynomial for Pretzel Links

In this paper, we deal with basic properties of some pretzel links and properties of the Jones polynomials of some pretzel links. By using these properties, the zero distribution of pretzel links is st...     

The polynomial invariants of twisted links

A twisted link is a generalization of a virtual link, which is related to a link diagram on a closed, possibly non-orientable surface. In this paper we generalize the Miyazawa polynomial invariant of a virtual link to an invariant of a twisted link in two formulae one of which is introduced by A. Ishii and the other by the author.     

Vassiliev invariants of type two for a link

We show that any type two Vassiliev invariant of a link can be expressed as a linear combination of the second coefficients of the Conway polynomials of its components and a quadratic expression of linking numbers.