Narrow Results

Let $X$ be a complex smooth quasi-projective variety with a fixed epimorphism $\nu :{\pi }_1(X)\twoheadrightarrow \mathbb{Z}$. We consider the asymptotic behavior of invariants such as Betti numbers with any field coefficients and the order of the torsion part of singular integral homology associated to $\nu$, known as the $L^2$-type invariants. At homological degree one, we give concrete formulas for these limits by the geometric information of $X$ when $\nu$ is orbifold effective. The proof relies on a study about cohomological degree one jump loci of $X$. We extend part of Arapura’s result for cohomological degree one jump loci of $X$ with complex field coefficients to the one with positive characteristic field coefficients. As an application, when $X$ is a hyperplane arrangement complement, a combinatoric upper bound is given for the number of parallel positive-dimensional components in cohomological degree one jump loci with complex coefficients. Another application is that we give a positive answer to a question posed by Denham and Suciu for hyperplane arrangement.

In this paper, we give a classification of Alexander modules of null-homologous knots in rational homology spheres. We characterize these modules equipped with their Blanchfield forms ϕ, and the modules such that there is a unique isomorphism class of , and we prove that for the other modules , there are infinitely many such classes. We realize all these by explicit knots in ℚ-spheres.

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.

We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module $M$ over the Laurent polynomial ring ${\mathrm{\Lambda }}_{\mu }=\mathbb{Z}[t_1^{\pm 1},\dots ,t_{\mu }^{\pm 1}]$. If $D$ is a diagram of a link $L$ with $\mu$ components, then the colorings of $D$ with values in $M$ form a ${\mathrm{\Lambda }}_{\mu }$-module ${\mathrm{\text{Color}}}_A(D,M)$. Extending a result of Inoue [Knot quandles and infinite cyclic covering spaces, Kodai Math. J.33 (2010) 116–122], we show that ${\mathrm{\text{Color}}}_A(D,M)$ is isomorphic to the module of ${\mathrm{\Lambda }}_{\mu }$-linear maps from the Alexander module of $L$ to $M$. In particular, suppose $M$ is a field and $\phi :{\mathrm{\Lambda }}_{\mu }\to M$ is a homomorphism of rings with unity. Then $\phi$ defines a ${\mathrm{\Lambda }}_{\mu }$-module structure on $M$, which we denote $M_{\phi }$. We show that the dimension of ${\mathrm{\text{Color}}}_A(D,M_{\phi })$ as a vector space over $M$ is determined by the images under $\phi$ of the elementary ideals of $L$. This result applies in the special case of Fox tricolorings, which correspond to $M=\mathrm{\text{GF}}(3)$ and $\phi (t_i)\equiv -1$. Examples show that even in this special case, the higher Alexander polynomials do not suffice to determine $|{\mathrm{\text{Color}}}_A(D,M_{\phi })|$; this observation corrects erroneous statements of Inoue [Quandle homomorphisms of knot quandles to Alexander quandles, J. Knot Theory Ramifications10 (2001) 813–821; op. cit.].

If $L$ is a classical link then the multivariate Alexander quandle, $Q_A(L)$, is a substructure of the multivariate Alexander module, $M_A(L)$. In the first paper of this series, we showed that if two links $L$ and $L^{\prime }$ have $Q_A(L)\cong Q_A(L^{\prime })$, then after an appropriate re-indexing of the components of $L$ and $L^{\prime }$, there will be a module isomorphism $M_A(L)\cong M_A(L^{\prime })$ of a particular type, which we call a “Crowell equivalence.” In this paper, we show that $Q_A(L)$ (up to quandle isomorphism) is a strictly stronger link invariant than $M_A(L)$ (up to re-indexing and Crowell equivalence). This result follows from the fact that $Q_A(L)$ determines the $Q_A$ quandles of all the sublinks of $L$, up to quandle isomorphisms.

The multivariate Alexander module of a link $L$ has several subsets that admit quandle operations defined using the module operations. One of them, the fundamental multivariate Alexander quandle, determines the link module sequence of $L$.

Joyce showed that for a classical knot $K$, the order of the involutory medial quandle is $\left|\mathrm{\text{det}}K\right|$. Generalizing Joyce’s result, we show that for a classical link $L$ of $\mu \ge 1$ components, the order of the involutory medial quandle is $\mu \left|\mathrm{\text{det}}L\right|/2^{\mu -1}$. In particular, $\mathrm{\text{IMQ}}(L)$ is infinite if and only if $detL=0$. We also relate $\mathrm{\text{IMQ}}(L)$ to several other link invariants.

Joyce showed that for a classical knot $K$, the involutory medial quandle $\mathrm{\text{IMQ}}(K)$ is isomorphic to the core quandle of the homology group $H_1(X_2)$, where $X_2$ is the cyclic double cover of ${𝕊}^3$, branched over $K$. It follows that $|\mathrm{\text{IMQ}}(K)|=|detK|$. In this paper, the extension of Joyce’s result to classical links is discussed. Among other things, we show that for a classical link $L$ of $\mu \ge 2$ components, the order of the involutory medial quandle is bounded as follows:

We discuss meridians and longitudes in reduced Alexander modules of classical and virtual links. When these elements are suitably defined, each link component will have many meridians, but only one longitude. Enhancing the reduced Alexander module by singling out these peripheral elements provides a significantly stronger link invariant. In particular, the enhanced module determines all linking numbers in a link; in contrast, the module alone does not even detect how many linking numbers are 0.

Joyce observed that the Alexander invariant and the medial quandle of a classical knot are equivalent to each other, as invariants. In this paper, we discuss the rather complicated extension of Joyce’s observation to several different medial quandles and reduced (one-variable) Alexander modules associated with classical links. The theme is that for links, medial quandles provide stronger invariants than reduced Alexander modules.

In this paper, we explain how the medial quandle of a classical or virtual link can be built from the peripheral structure of the reduced Alexander module.