Fibred knots and twisted Alexander invariants

We study the twisted Alexander invariants of fibred knots. We establish necessary conditions on the twisted Alexander invariants for a knot to be fibred, and develop a practical method to compute the twisted Alexander invariants from the homotopy type of a monodromy. It is illustrated that the twisted Alexander invariants carry more information on fibredness than the classical Alexander invariants, even for knots with trivial Alexander polynomials.

     

Multivariable Alexander invariants of hypersurface complements

We start with a discussion on Alexander invariants, and then prove some general results concerning the divisibility of the Alexander polynomials and the supports of the Alexander modules, via Artin's vanishing theorem for perverse sheaves. We conclude with explicit computations of twisted cohomology following an idea already exploited in the hyperplane arrangement case, which combines the degeneration of the Hodge to de Rham spectral sequence with the purity of some cohomology groups.

     

Intersection Alexander polynomials

By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the intersection Alexander polynomials satisfy certain duality and normalization conditions analogous to those of ordinary Alexander polynomials, and we explore the relationships between the intersection Alexander polynomials and certain generalizations of the classical Alexander polynomials that are defined for non-locally-flat knots. We also investigate the relations between the intersection Alexander polynomials of a knot and the intersection and classical Alexander polynomials of the link knots around the singular strata. To facilitate some of these investigations, we introduce spectral sequences for the computation of the intersection homology of certain stratified bundles.     

Representations of Knot Groups and Twisted Alexander Polynomials

We present a twisted version of the Alexander polynomial associated with a matrix representation of the knot group. Examples of two knots with the same Alexander module but different twisted Alexander polynomials are given.     

The Thurston norm, fibered manifolds and twisted Alexander polynomials

Every element in the first cohomology group of a 3-manifold is dual to embedded surfaces. The Thurston norm measures the minimal ‘complexity’ of such surfaces. For instance the Thurston norm of a knot complement determines the genus of the knot in the 3-sphere. We show that the degrees of twisted Alexander polynomials give lower bounds on the Thurston norm, generalizing work of McMullen and Turaev. Our bounds attain their most concise form when interpreted as the degrees of the Reidemeister torsion of a certain twisted chain complex. We show that these lower bounds give the correct genus bounds for all knots with 12 crossings or less, including the Conway knot and the Kinoshita-Terasaka knot which have trivial Alexander polynomial.We also give obstructions to fibering 3-manifolds using twisted Alexander polynomials and detect all knots with 12 crossings or less that are not fibered. For some of these it was unknown whether or not they are fibered. Our work in particular extends the fibering obstructions of Cha to the case of closed manifolds.     

Knot theory for self-indexed graphs

We introduce and study so-called self-indexed graphs. These are (oriented) finite graphs endowed with a map from the set of edges to the set of vertices. Such graphs naturally arise from classical knot and link diagrams. In fact, the graphs resulting from link diagrams have an additional structure, an integral flow. We call a self-indexed graph with integral flow a comte. The analogy with links allows us to define transformations of comtes generalizing the Reidemeister moves on link diagrams. We show that many invariants of links can be generalized to comtes, most notably the linking number, the Alexander polynomials, the link group, etc. We also discuss finite type invariants and quandle cocycle invariants of comtes.

     

On Invariants of Virtual Links

We propose a new method of generalizing classical link invariants for the case of virtual links. In particular, we have generalized the knot quandle, the knot fundamental group, the Alexander module, and the coloring invariants. The virtual Alexander module leads to a definition of VA-polynomial that has no analogue in the classical case (i.e. vanishes on classical links).     

Differences of Alexander polynomials for knots caused by a single crossing change

Kondo and Sakai independently gave a characterization of Alexander polynomials for knots which are transformed into the trivial knot by a single crossing change. The first author gave a characterization of Alexander polynomials for knots which are transformed into the trefoil knot (and into the figure-eight knot) by a single crossing change. In this note, we will give a characterization of Alexander polynomials for knots which are transformed into the 10₁₃₂ knot (and into the (5,2)-torus knot) by a single crossing change. Moreover, this method can be applied for knots with monic Alexander polynomials.     

On the Alexander polynomials of knots with Gordian distance one

We consider a condition on a pair of the Alexander polynomials of knots which are realizable by a pair of knots with Gordian distance one. We show that there are infinitely many mutually disjoint infinite subsets in the set of the Alexander polynomials of knots such that every pair of distinct elements in each subset is not realizable by any pair of knots with Gordian distance one. As one of the subsets, we have an infinite set containing the Alexander polynomials of the trefoil knot and the figure eight knot. We also show that every pair of distinct Alexander polynomials such that one is the Alexander polynomial of a slice knot is realizable by a pair of knots of Gordian distance one, so that every pair of distinct elements in the infinite subset consisting of the Alexander polynomials of slice knots is realizable by a pair of knots with Gordian distance one. These results solve problems given by Y. Nakanishi and by I. Jong.     

Non vanishing loci of Hodge numbers of local systems

We show that closures of families of unitary local systems on quasiprojective varieties for which the dimension of a graded component of Hodge filtration has a constant value can be identified with a finite union of polytopes. We also present a local version of this theorem. This yields the “Hodge decomposition” of the set of unitary local systems with a non-vanishing cohomology extending Hodge decomposition of characteristic varieties of links of plane curves studied by the author earlier. We consider a twisted version of the characteristic varieties generalizing the twisted Alexander polynomials. Several explicit calculations for complements to arrangements are made. A. Libgober was supported by National Science Foundation grant.     

Alexander polynomials of alternating knots of genus two III

In the previous paper, the author gave linear inequalities on the coefficients of the Alexander polynomials of alternating knots of genus two, which are best possible as linear inequalities on the coefficients of them. In this paper, we give infinitely many Alexander polynomials which satisfy the linear inequalities, but they are not realized by alternating knots.     

Twisted Alexander norms give lower bounds on the Thurston norm

We introduce twisted Alexander norms of a compact connected orientable 3-manifold with first Betti number greater than one, generalizing norms of McMullen and Turaev. We show that twisted Alexander norms give lower bounds on the Thurston norm of a 3-manifold. Using these we completely determine the Thurston norm of many 3-manifolds which are not determined by norms of McMullen and Turaev.

     

Yang–Baxter Operators Arising from Algebra Structures and the Alexander Polynomial of Knots

ABSTRACT

In this paper, we consider the problem of constructing knot invariants from Yang–Baxter operators associated to algebra structures. We first compute the enhancements of these operators. Then, we conclude that Turaev's procedure to derive knot invariants from these enhanced operators, as modified by Murakami, invariably produces the Alexander polynomial of knots.     

Chamber Structure for Some Equivariant Relative Gromov–Witten Invariants of ℙ<Superscript>1</Superscript> in Genus 0

In this paper, we study genus 0 equivariant relative Gromov-Witten invariants of P¹ whose corresponding relative stable maps are totally ramified over one point. For fixed number of marked points, we show that such invariants are piecewise polynomials in some parameter space. The parameter space can then be divided into polynomial domains, called chambers. We determine the difference of polynomials between two neighbouring chambers. In some special chamber, which we called the totally negative chamber, we show that such a polynomial can be expressed in a simple way. The chamber structure here shares some similarities to that of double Hurwitz numbers.     

Polynomial invariants and quandles of twisted links

Bourgoin defined the notion of a twisted link which corresponds to a stable equivalence class of links in oriented thickenings. It is a generalization of a virtual link. Some invariants of virtual links are extended for twisted links including the knot group and the Jones polynomial. In this paper, we generalize a multivariable polynomial invariant of a virtual link to a twisted link. We also introduce a quandle of a twisted link.     

Gerbes and twisted orbifold quantum cohomology

In this paper,we construct an orbifold quantum cohomology twisted by a flat gerbe. Then we compute these invariants in the case of a smooth manifold and a discrete torsion on a global quotient orbifold.     

Alexander polynomials of equivariant slice and ribbon knots in

This paper gives an algebraic characterization of Alexander polynomials of equivariant ribbon knots and a factorization condition satisfied by Alexander polynomials of equivariant slice knots.

     

Triple point cancelling numbers of surface links and quandle cocycle invariants

The unknotting or triple point cancelling number of a surface link F is the least number of 1-handles for F such that the 2-knot obtained from F by surgery along them is unknotted or pseudo-ribbon, respectively. These numbers have been often studied by knot groups and Alexander invariants. On the other hand, quandle colorings and quandle cocycle invariants of surface links were introduced and applied to other aspects, including non-invertibility and triple point numbers. In this paper, we give lower bounds of the unknotting or triple point cancelling numbers of surface links by using quandle colorings and quandle cocycle invariants.