Narrow Results

The axioms of a quandle imply that the columns of its Cayley table are permutations. This paper studies quandles with exactly one non-trivially permuted column. Their automorphism groups, quandle polynomials, (symmetric) cohomology groups, and $\mathrm{\text{Hom}}$ quandles are studied. The quiver and cocycle invariant of links using these quandles are shown to relate to linking number.

As a generalization of quandles, biquandles have given many invariants of classical/surface/virtual links. In this paper, we show that the fundamental quandle $Q(L)$ of any classical/surface link $L$ detects the fundamental biquandle $\mathrm{\text{BQ}}(L)$; more precisely, there exists a functor ${ℬ}$ from the category of quandles to that of biquandles such that ${ℬ}(Q(L))\cong \mathrm{\text{BQ}}(L)$. Then, we can expect invariants from biquandles to be reduced to those from quandles. In fact, we introduce a right-adjoint functor ${𝒬}$ of ${ℬ}$, which implies that the coloring number of a biquandle $X$ is equal to that of the quandle ${𝒬}(X)$.

Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and is extended to links component-wise.

We resume work on telling embeddings of codimension two apart by counting colorings of the corresponding diagrams by given quandles. Previously, we illustrated the efficiency of this approach on classical knots. In the present paper we apply it to knotted surfaces. We recover work of Kamada in telling ribbon knots apart and we distinguish all elements of a class of twist-spun torus knots.

We construct a series of combinatorial quandle-like knot invariants. We color regions of a knot diagram rather than lines and assign a weight to each coloring. Sets of these weights are the invariants we construct (colorings and weights depend on several parameters).

Using these invariants, we prove that left and right trefoils are not isotopic using this invariant (in a particular case).

We consider several approaches to defining a link homotopy version of the fundamental quandle Q(L) of a link L in S³. We first define the reduced fundamental quandle RQ(L) as a quotient of Q(L). We show that RQ(L) is a link homotopy invariant that carries at least as much information as the meridian-preserving isomorphism class of Milnor's reduced group RG(L). We then show that operator reduction, a plausible alternative approach to defining RQ(L), fails to yield a link homotopy invariant. Finally, we give a geometric characterization of RQ(L), and offer a caveat regarding a seemingly simpler approach.

A rack shadow or rack set is a set X with a rack action by a rack R, analogous to a vector space over a field. We use shadow colorings of classical link diagrams to define enhanced rack counting invariants and show that the enhanced invariants are stronger than unenhanced counting invariants.

The classical trefoil is famous for having a 3-coloring which distinguishes it from the unknot. The 3-coloring is also notorious for not distinguishing the right-handed from the left-handed trefoil. However with a bit of tweaking the three colors can also be used for this task. What lies behind the method is an extension of the definition of a birack to a partial birack. An operation on biracks called doubling converts the 3-color quandle into a partial biquandle. Coloring with this biquandle distinguishes the right-handed from the left-handed trefoil. Equivalently it defines an element of the homology of the quandle or biquandle classifying space. The same treatment is applied to 2-coloring a diagram of a knot. A useful spin-off is a simplified and extended treatment of biracks and biquandles. Full details are given.

Using the classification of transitive groups we classify indecomposable quandles of size < 36. This classification is available in Rig, a GAP package for computations related to racks and quandles. As an application, the list of all indecomposable quandles of size < 36 not of type D is computed.

This paper gives an alternate definition of the Affine Index Polynomial (called the Wriggle Polynomial) using virtual linking numbers and explores applications of this polynomial. In particular, it proves the Cosmetic Crossing Change Conjecture for odd virtual knots and pure virtual knots. It also demonstrates that the polynomial can detect mutations by positive rotation and proves it cannot detect mutations by positive reflection. Finally it exhibits a pair of mutant knots that can be distinguished by a type 2 vassiliev invariant coming from the polynomial.

If A is an abelian quandle and Q is a quandle, the hom set Hom(Q, A) of quandle homomorphisms from Q to A has a natural quandle structure. We exploit this fact to enhance the quandle counting invariant, providing an example of links with the same counting invariant values but distinguished by the hom quandle structure. We generalize the result to the case of biquandles, collect observations and results about abelian quandles and the hom quandle, and show that the category of abelian quandles is symmetric monoidal closed.

We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of 23 finite quandles. We study the colorings of these 2977 knots by all of the 431 connected quandles of order at most 35 found by Vendramin. Among other things, we collect information about quandles that have the same number of colorings for all of the 2977 knots. For example, we prove that if Q is a simple quandle of prime power order then Q and the dual quandle Q* of Q have the same number of colorings for all knots and conjecture that this holds for all Alexander quandles Q. We study a knot invariant based on a quandle homomorphism f : Q₁ → Q₀. We also apply the quandle colorings we have computed to obtain some new results for the bridge index, the Nakanishi index, the tunnel number, and the unknotting number. In an appendix we discuss various properties of the quandles in Vendramin's list. Links to the data computed and various programs in C, GAP and Maple are provided.

Quandle cocycle invariants form a powerful and well-developed tool in knot theory. This paper treats their variations — namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow world are extended to the positive and twisted cases. Another application is a sharpening of twisted quandle cocycle invariants for multi-component links.

Foams in all dimensions are defined by being modeled on a dual structure found in an $n$-dimensional simplex. Their crossings are studied from geometric and homological points of view. A process of defining invariants thereof is outlined. Interesting research level problems are proposed. From considerations of foams, a partial order on certain families of forests is given via Hasse diagrams.

We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then introduced providing a first step toward a general continuous cohomology theory for topological racks and quandles.

Quandle 2-cocycles define invariants of classical and virtual knots, and extensions of quandles. We show that the quandle 2-cocycle invariant with respect to a non-trivial $2$-cocycle is constant, or takes some other restricted form, for classical knots when the corresponding extensions satisfy certain algebraic conditions. In particular, if an abelian extension is a conjugation quandle, then the corresponding cocycle invariant is constant. Specific examples are presented from the list of connected quandles of order less than 48. Relations among various quandle epimorphisms involved are also examined.

We explore a knot invariant derived from colorings of corresponding $1$-tangles with arbitrary connected quandles. When the quandle is an abelian extension of a certain type the invariant is equivalent to the quandle $2$-cocycle invariant. We construct many such abelian extensions using generalized Alexander quandles without explicitly finding $2$-cocycles. This permits the construction of many $2$-cocycle invariants without exhibiting explicit $2$-cocycles. We show that for connected generalized Alexander quandles the invariant is equivalent to Eisermann’s knot coloring polynomial. Computations using this technique show that the $2$-cocycle invariant distinguishes all of the oriented prime knots up to 11 crossings and most oriented prime knots with 12 crossings including classification by symmetry: mirror images, reversals, and reversed mirrors.

The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular links. As an application, we distinguish several singular knots and links.

In the study of knotted trivalent graphs and their higher dimensional analogue, knotted foams, some of the moves have alternative interpretations. Here three interpretations are given. (1) As the boundaries of chains in a homology theory, (2) as a system of abstract tensor relations, and (3) as a collection of polyhedra that include the permutohedron. The homological interpretation will allow for a solution to the abstract tensor system.

We give a generating set of the generalized Reidemeister moves for oriented singular links. We then introduce an algebraic structure arising from the axiomatization of Reidemeister moves on oriented singular knots. We give some examples, including some non-isomorphic families of such structures over non-abelian groups. We show that the set of colorings of a singular knot by this new structure is an invariant of oriented singular knots and use it to distinguish some singular links.