Alexander Module Research Articles

Twisted Alexander invariants of complex hypersurface complements

We define and study twisted Alexander-type invariants of complex hypersurface complements. We investigate torsion properties for the twisted Alexander modules and extend the local-to-global divisibility results of Maxim and of Dimca and Libgober to the twisted setting. In the process, we also study the splitting fields containing the roots of the corresponding twisted Alexander polynomials.

A Burau–Alexander 2-functor on tangles

We construct a weak 2-functor from the bicategory of oriented tangles to a bicategory of Lagrangian cospans. This functor simultaneously extends the Burau representation of the braid groups, its generalization to tangles due to Turaev and the first-named author, and the Alexander module of 1 and 2-dimensional links.

Twisted Alexander Invariants Detect Trivial Links

AbstractIt follows fromearlier work of Silver andWilliams and the authors that twisted Alexander polynomials detect the unknot and theHopf link. We now show that twisted Alexander polynomials also detect the trefoil and the ûgure-Ç knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexander modules detect trivial links. We use this result to provide algorithms for detecting whether a link is the unlink, whether it is split, and whether it is totally split.

Cohomology support loci of local systems

The support S of Sabbah's specialization complex is a simultaneous generalization of the set of eigenvalues of the monodromy on Deligne's nearby cycles complex, of the support of the Alexander modules of an algebraic knot, and of certain cohomology support loci. Moreover, it equals conjecturally the image under the exponential map of the zero locus of the Bernstein-Sato ideal. Sabbah showed that S is contained in a union of translated subtori of codimension one in a complex affine torus. Budur-Wang showed recently that S is a union of torsion-translated subtori. We show here that S is always a hypersurface, and that it admits a formula in terms of log resolutions. As an application, we give a criterion in terms of log resolutions for the (semi-)simplicity as perverse sheaves, or as regular holonomic D-modules, of the direct images of rank one local systems under an open embedding. For hyperplane arrangements, this criterion is combinatorial.

Spectral pairs, Alexander modules, and boundary manifolds

Let \(n>0\) and \(f: {\mathbb {C}}^{n+1}\rightarrow {\mathbb {C}}\) be a reduced polynomial map, with \(D=f^{-1}(0)\), \({\mathcal {U}}={\mathbb {C}}^{n+1}{\setminus } D\) and boundary manifold \(M=\partial {\mathcal {U}}\). Assume that f is transversal at infinity and D has only isolated singularities. Then the only interesting non-trivial Alexander modules of \({\mathcal {U}}\) and resp. M appear in the middle degree n. We revisit the mixed Hodge structures on these Alexander modules and study their associated spectral pairs (or equivariant mixed Hodge numbers). We obtain upper bounds for the spectral pairs of the n-th Alexander module of \({\mathcal {U}}\), which can be viewed as a Hodge-theoretic refinement of Libgober’s divisibility result for the corresponding Alexander polynomials. For the boundary manifold M, we show that the spectral pairs associated to the non-unipotent part of the n-th Alexander module of M can be computed in terms of local contributions (coming from the singularities of D) and contributions from “infinity”.

Twisted cohomology pairings of knots I; diagrammatic computation

We provide a diagrammatic computation for the bilinear form, which is defined as the pairing between the (relative) cup products with every local coefficients and every integral homology 2-class of every link in the 3-sphere. As a corollary, we construct bilinear forms on the twisted Alexander modules of links.

A RECURSIVE FORMULA FOR THE KHOVANOV COHOMOLOGY OF KANENOBU KNOTS

Kanenobu has given infinite families of knots with the same HOMFLY polynomial invariant but distinct Alexander module structure. In this paper, we give a recursive formula for the Khovanov cohomology of all Kanenobu knots K(p, q), where p and q are integers. The result implies that the rank of the Khovanov cohomology of K(p, q) is an invariant of p + q. Our computation uses only the basic long exact sequence in knot homology and some results on homologically thin knots.

Equivariant triple intersections

Given a null-homologous knot $K$ in a rational homology 3-sphere $M$, and the standard infinite cyclic covering $\tilde{X}$ of $(M,K)$, we define an invariant of triples of curves in $\tilde{X}$, by means of equivariant triple intersections of surfaces. We prove that this invariant provides a map $\phi$ on $\Al^{\otimes 3}$, where $\Al$ is the Alexander module of $(M,K)$, and that the isomorphism class of $\phi$ is an invariant of the pair $(M,K)$. For a fixed Blanchfield module $(\Al,\bl)$, we consider pairs $(M,K)$ whose Blanchfield modules are isomorphic to $(\Al,\bl)$, equipped with a marking, {\em i.e.} a fixed isomorphism from $(\Al,\bl)$ to the Blanchfield module of $(M,K)$. In this setting, we compute the variation of $\phi$ under null borromean surgeries, and we describe the set of all maps $\phi$. Finally, we prove that the map $\phi$ is a finite type invariant of degree 1 of marked pairs $(M,K)$ with respect to null Lagrangian-preserving surgeries, and we determine the space of all degree 1 invariants of marked pairs $(M,K)$ with rational values.

The Alexander module, Seifert forms, and categorification

We show that bordered Floer homology provides a categorification of a TQFT described by Donaldson. This, in turn, leads to a proof that both the Alexander module of a knot and the Seifert form are completely determined by Heegaard Floer theory.

Characteristic varieties of hypersurface complements

We give divisibility results for the (global) characteristic varieties of hypersurface complements expressed in terms of the local characteristic varieties at points along one of the irreducible components of the hypersurface. As an application, we recast old and obtain new finiteness and divisibility results for the classical (infinite cyclic) Alexander modules of complex hypersurface complements. Moreover, for the special case of hyperplane arrangements, we translate our divisibility results for characteristic varieties in terms of the corresponding resonance varieties.

On the Néron–Severi lattice of a Delsarte surface

We suggest an algorithm computing, in some cases, an explicit generating set for the Néron–Severi lattice of a Delsarte surface. In a few special cases, including those of Fermat surfaces and cyclic Delsarte surfaces that were previously conjectured in the literature, we show that certain “obvious” divisors do generate the lattice. The proof is based on the computation of the Alexander module related to a certain abelian covering.

Blanchfield forms and Gordian distance

Given a link in S3S3 we will use invariants derived from the Alexander module and the Blanchfield pairing to obtain lower bounds on the Gordian distance between links, the unlinking number and various splitting numbers. These lower bounds generalise results recently obtained by Kawauchi. We give an application restricting the knot types which can arise from a sequence of splitting operations on a link. This allows us to answer a question asked by Colin Adams in 1996.

Classical homological invariants are not determined by knot Floer homology and Khovanov homology

We illustrate that there are knots for which Heegaard knot Floer homology and Khovanov homology are identical but the Alexander module and torsion invariants differ. The examples are certain symmetric unions. We also give examples of similar flavor, concerning the Kauffman and Q-polynomials in place of the classical homological invariants. This shows there are nonmutant knots with the same knot Floer and Khovanov homology.

Colorings, determinants and Alexander polynomials for spatial graphs

A balanced spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to S. Kinoshita, Alexander polynomials as isotopy invariants I, Osaka Math. J. 10 (1958) 263–271.), and examine their behavior under some common operations on the graph. We use the Alexander module to define the determinant and [Formula: see text]-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which [Formula: see text] the graph is [Formula: see text]-colorable, and that a [Formula: see text]-coloring of a graph corresponds to a representation of the fundamental group of its complement into a metacyclic group [Formula: see text]. We finish by proving some properties of the Alexander polynomial.

Nearby cycles and Alexander modules of hypersurface complements

Let f:Cn+1→C be a polynomial map, which is transversal at infinity. Using Sabbah's specialization complex, we give a new description of the Alexander modules of the hypersurface complement Cn+1∖f−1(0), and obtain a general divisibility result for the associated Alexander polynomials. As a byproduct, we prove a conjecture of Maxim on the decomposition of the Cappell–Shaneson peripheral complex of the hypersurface. Moreover, as an application, we use nearby cycles to recover the mixed Hodge structure on the torsion Alexander modules, as defined by Dimca and Libgober. We also explore the relation between the generic fibre of f and the nearby cycles.

Reidemeister torsion, peripheral complex and Alexander polynomials of hypersurface complements

Let $f:\CN \rightarrow \C $ be a polynomial, which is transversal (or regular) at infinity. Let $\U=\CN\setminus f^{-1}(0)$ be the corresponding affine hypersurface complement. By using the peripheral complex associated to $f$, we give several estimates for the (infinite cyclic) Alexander polynomials of $\U$ induced by $f$, and we describe the error terms for such estimates. The obtained polynomial identities can be further refined by using the Reidemeister torsion, generalizing a similar formula proved by Cogolludo and Florens in the case of plane curves. We also show that the above-mentioned peripheral complex underlies an algebraic mixed Hodge module. This fact allows us to construct mixed Hodge structures on the Alexander modules of the boundary manifold of $\U$.

Rational Blanchfield forms, S-equivalence, and null LP-surgeries

Null Lagrangian-preserving surgeries are a generalization of the Garoufalidis and Rozansky null-moves, that these authors introduced to study the Kricker lift of the Kontsevich integral, in the setting of pairs (M,K) composed of a rational homology sphere M and a null-homologous knot K in M. They are defined as replacements of null-homologous rational homology handlebodies of M\K by other such handlebodies with identical Lagrangian. A null Lagrangian-preserving surgery induces a canonical isomorphism between the rational Alexander modules of the involved pairs, which preserves the Blanchfield form. Conversely, we prove that a fixed isomorphism between rational Alexander modules which preserves the Blanchfield form can be realized, up to multiplication by a power of t, by a finite sequence of null Lagrangian-preserving surgeries. We also prove that such classes of isomorphisms can be realized by rational S-equivalences. In the case of integral homology spheres, we prove similar realization results for a fixed isomorphism between integral Alexander modules.

REVISIT TO CONNECTED ALEXANDER QUANDLES OF SMALL ORDERS VIA FIXED POINT FREE AUTOMORPHISMS OF FINITE ABELIAN GROUPS

In this paper we provide a rigorous proof for the fact that there are exactly 8 connected Alexander quandles of order <TEX>$2^5$</TEX> by combining properties of fixed point free automorphisms of finite abelian 2-groups and the classification of conjugacy classes of GL(5, 2). Furthermore we verify that six of the eight associated Alexander modules are simple, whereas the other two are semisimple.

On Computing the First Higher-Order Alexander Modules of Knots

Cochran defined the nth-order integral Alexander module of a knot in the three-sphere as the first homology group of the knot’s (n + 1)st iterated abelian cover. The case n = 0 gives the classical Alexander module (and polynomial). After a localization, one can obtain a finitely presented module over a principal ideal domain, from which one can extract a higher-order Alexander polynomial. We present an algorithm to compute the first-order Alexander module for any knot. As applications, we show that these higher-order Alexander polynomials provide a better bound on the knot genus than does the classical Alexander polynomial, and that they distinguish mutant knots. Included in this algorithm is a solution to the word problem in finitely presented -modules.

The Alexander module of a trigonal curve

We describe the Alexander modules and Alexander polynomials (both over \mathbb{Q} and over finite fields \mathbb{F}_p ) of generalized trigonal curves. The rational case is completely resolved; in the case of characteristic p&gt;0 , a few points remain open. The results obtained apply as well to plane curves with deep singularities.