Multivariate Alexander quandles, II. The involutory medial quandle of a link (corrected)

Joyce showed that for a classical knot [Formula: see text], the involutory medial quandle [Formula: see text] is isomorphic to the core quandle of the homology group [Formula: see text], where [Formula: see text] is the cyclic double cover of [Formula: see text], branched over [Formula: see text]. It follows that [Formula: see text]. In this paper, the extension of Joyce’s result to classical links is discussed. Among other things, we show that for a classical link [Formula: see text] of [Formula: see text] components, the order of the involutory medial quandle is bounded as follows: [Formula: see text] In particular, [Formula: see text] is infinite if and only if [Formula: see text]. We also show that in general, [Formula: see text] is a strictly stronger invariant than [Formula: see text]. That is, if [Formula: see text] and [Formula: see text] are links with [Formula: see text], then [Formula: see text]; but it is possible to have [Formula: see text] and [Formula: see text]. In fact, it is possible to have [Formula: see text] and [Formula: see text].

Multivariate Alexander quandles, II. The involutory medial quandle of a link

Joyce showed that for a classical knot [Formula: see text], the order of the involutory medial quandle is [Formula: see text]. Generalizing Joyce’s result, we show that for a classical link [Formula: see text] of [Formula: see text] components, the order of the involutory medial quandle is [Formula: see text]. In particular, [Formula: see text] is infinite if and only if [Formula: see text]. We also relate [Formula: see text] to several other link invariants.

Multivariate Alexander quandles, III. Sublinks

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

A note on Dehn colorings and invariant factors

If [Formula: see text] is an abelian group and [Formula: see text] is an integer, let [Formula: see text] be the subgroup of [Formula: see text] consisting of elements [Formula: see text] such that [Formula: see text]. We prove that if [Formula: see text] is a diagram of a classical link [Formula: see text] and [Formula: see text] are the invariant factors of an adjusted Goeritz matrix of [Formula: see text], then the group [Formula: see text] of Dehn colorings of [Formula: see text] with values in [Formula: see text] is isomorphic to the direct product of [Formula: see text] and [Formula: see text]. It follows that the Dehn coloring groups of [Formula: see text] are isomorphic to those of a connected sum of torus links [Formula: see text].

Linking invariants of even virtual links

A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible.

Up–down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type II

We introduce an up–down coloring of a virtual-link (or classical-link) diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two [Formula: see text]-component virtual-link (or classical-link) diagrams. By using the notion of a quandle cocycle invariant, we give a method to detect the necessity of Reidemeister moves of type II between two given virtual-knot (or classical-knot) diagrams. As an application, we show that for any virtual-knot diagram [Formula: see text], there exists a diagram [Formula: see text] representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister move of type II.

Classical link invariants from the framizations of the Iwahori–Hecke algebra and the Temperley–Lieb algebra of type A

In this paper, we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma–Hecke algebras [Formula: see text], which are not topologically equivalent to the Homflypt polynomial. We then present the algebra [Formula: see text] which is the appropriate Temperley–Lieb analogue of [Formula: see text], as well as the related 1-variable classical link invariants, which in turn are not topologically equivalent to the Jones polynomial. Finally, we present the algebra of braids and ties which is related to the Yokonuma–Hecke algebra, and also its quotient, the partition Temperley–Lieb algebra [Formula: see text] and we prove an isomorphism of this algebra with a subalgebra of [Formula: see text].

Extended quandle spaces and shadow homotopy invariants of classical links

In 1993, Fenn, Rourke and Sanderson introduced rack spaces and rack homotopy invariants, and modifications to quandle spaces and quandle homotopy invariants were introduced by Nosaka in 2011. In this paper, we define the Cayley-type graph and the extended quandle space of a quandle in analogy to rack and quandle spaces. Moreover, we construct the shadow homotopy invariant of a classical link and prove that the shadow homotopy invariant is equal to the quandle homotopy invariant multiplied by the order of a quandle.

Applying Lipson's state models to marked graph diagrams of surface-links

A. S. Lipson constructed two state models yielding the same classical link invariant obtained from the Kauffman polynomial F(a, u). In this paper, we apply Lipson's state models to marked graph diagrams of surface-links, and observe when they induce surface-link invariants.

Links and Planar Diagram Codes

In this paper we formalize a combinatorial object for describing link diagrams called a Planar Diagram Code (PD-Code). PD-codes are used by the KnotTheory Mathematica package developed by Bar-Natan et al. We present the set of PD-codes as a standalone object and discuss its relationship with link diagrams. We give an explicit algorithm for reconstructing a knot diagram on a surface from a PD-code. We also discuss the intrinsic symmetries of PD-codes (i.e. invertibility and chirality). The moves analogous to the Reidemeister moves are also explored, and we show that the given set of PD-codes modulo these combinatorial Reidemeister moves is equivalent to classical link theory.