Narrow Results

In this paper, we generalize virtual knot theory to multi-virtual knot theory where there are a multiplicity of virtual crossings. Each virtual crossing type can detour over the other virtual crossing types, and over classical or immersed crossings. New invariants of multi-virtual knots and links are introduced and new problems that arise are described. We show how the extensions of the Penrose coloring evaluation for trivalent plane graphs and our generalizations of this to non-planar graphs and arbitrary numbers of colors acts as a motivation for the construction of the multi-virtual theory.

The paper relates the 4-fold symmetric quandle homotopy (cocycle) invariants to topological objects. We show that the 4-fold symmetric quandle homotopy invariants are at least as powerful as the Dijkgraaf–Witten invariants. As an application, for an odd prime p, we show that the quandle cocycle invariant of a link in S³ constructed by the Mochizuki 3-cocycle is equivalent to the Dijkgraaf–Witten invariant with respect to ℤ/pℤ of the double covering of S³ branched along the link. We also reconstruct the Chern–Simons invariant of closed 3-manifolds as a quandle cocycle invariant via the extended Bloch group, in analogy to [A. Inoue and Y. Kabaya, Quandle homology and complex volume, preprint(2010), arXiv:math/1012.2923].

The purpose of this paper is to introduce and investigate the notion of derivation for quandle algebras. More precisely, we describe the symmetries on structure constants providing a characterization for a linear map to be a derivation. We obtain a complete characterization of derivations in the case of quandle algebras of dihedral quandles over fields of characteristic zero, and provide the dimensionality of the Lie algebra of derivations. Many explicit examples and computations are given over both zero and positive characteristic. Furthermore, we investigate inner derivations, in the sense of Schafer for non-associative structures. We obtain necessary conditions for the Lie transformation algebra of quandle algebras of Alexander quandles, with explicit computations in low dimensions.

A quandle is a set with a binary operation satisfying some properties. A quandle homomorphism is a map between quandles preserving the structure of their binary operations. A knot determines a quandle called a knot quandle. We show that the number of all quandle homomorphisms of a knot quandle of a knot to an Alexander quandle is completely determined by Alexander polynomials of the knot. Further we show that the set of all quandle homomorphisms of a knot quandle to an Alexander quandle has a module structure.

Given a tangle diagram of a classical knot K, we construct a surface diagram of any twist-spun 2-knot of K. From the crossings of the tangle diagram, we can get information of the corresponding triple points of the surface diagram, which are used to compute cocycle invariants of twist-spun 2-knots.

There are some phenomena arising in the virtual knot theory which are not the case for classical knots. One of them deals with the "breaking" procedure of knots and obtaining long knots. Unlike the classical case, they might not be the same. The present work is devoted to construction of some invariants of long virtual links. Several explicit examples are given. For instance, we show how to prove the non-triviality of some knots obtained by breaking virtual unknot diagrams by very simple means.

We prove that if a surface-knot diagram is colored by a specific quandle non-trivially, then any diagram of the surface-knot consists of at least four broken sheets. In particular, the minimal number of broken sheets for the spun trefoil is shown to be exactly four.

We calculate the dihedral quandle cocycle invariants of twist-spins of alternating odd pretzel knots. The calculation leads us to the conclusion that there exist non-ribbon 2-knots which admit a non-trivial coloring by the dihedral quandle R_p and all of whose cocycle invariants derived from ℤ_p-valued 3-cocycles on R_p take value in ℤ ⊂ ℤ[ℤ_p] for any odd prime integer p.

A left order on a magma (e.g. semigroup) is a total order of its elements that is left invariant under the magma operation. A natural topology can be introduced on the set of all left orders of an arbitrary magma. We prove that this topological space is compact. Interesting examples of nonassociative magmas, whose spaces of right orders we analyze, come from knot theory and are called quandles. Our main result establishes an interesting connection between topological properties of the space of left orders on a group, and the classical algebraic result by Conrad [4] and Łoś [13] concerning the existence of left orders.

Fox's shadow p-colorings of a knot K define two kinds of Laurent polynomials, Φ_p(K) and , as invariants of K. We prove that the equality holds for any knot K. Also we prove that, if the p-coloring number of K is equal to p², then has the form for some N ∈ ℤ_p.

This paper studies the biquandle as an invariant of virtual knots and links, analyzing its structure via composition of biquandle morphisms where elementary morphisms are associated with the crossings and virtual crossings in the diagram.

This paper summarizes substantive new results derived by a student team (the first three authors) under the direction of the fourth author at the 2005 session of the KSU REU "Brainstorming and Barnstorming". The main results are a decomposition theorem for quandles in terms of an operation of "semidisjoint union" showing that all finite quandles canonically decompose via iterated semidisjoint unions into connected subquandles, and a structure theorem for finite connected quandles with prescribed inner automorphism group. The latter theorem suggests a new approach to the classification of finite connected quandles.

We consider the problem of classification of links up to (2, 2)-moves. Our motivation comes from the theory of skein modules, more specifically from the skein module of S³ based on the deformation of (2, 2)-move. As it was proved in D-P-2, not every link can be reduced to a trivial link by (2, 2)-moves, for instance, the closure of (σ₁σ₂)⁶. In this paper, we classify 3-braids up to (2, 2)-moves and, we show how the Harikae–Nakanishi–Uchida conjecture can be modified to hold for closed 3-braids. As an important step in the classification we prove the conjecture for 2-algebraic links and classify (2, 2)-equivalence classes for links up to nine crossings. We also analyze an action of (2, 2)-move on Kei (involutive quandle) associated to a link diagram. We define Burnside Kei, Q(m, n), and ask for which values of m and n, is Q(m, n) finite. This question is motivated by classical Burnside problem.

We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of graph tangles, we define an invariant based on the walks in the graph and cocycles from nonabelian quandle cohomology.

This paper summarizes substantive new results derived by a student team (the first three authors) under the direction of the fourth author at the 2008 session of the KSU REU "Brainstorming and Barnstorming". The main results show that the construction of the inner automorphism group of a quandle gives rise to a functor from the category of quandles and surjective quandle homomorphisms to the category of groups, characterize quotient maps of quandles which do not change the group of inner automorphims, and characterize those normal subgroups of the inner automorphism group which arise as kernels of homomorphisms induced by quandle surjections.

In this paper we give the results of a computer search for biracks of small size and we give various interpretations of these findings. The list includes biquandles, racks and quandles together with new invariants of welded knots and examples of welded knots which are shown to be non-trivial by the new invariants. These can be used to answer various questions concerning virtual and welded knots. As an application we reprove the result that the Burau map from braids to matrices is non-injective and give an example of a non-trivial virtual (welded) knot which cannot be distinguished from the unknot by any linear biquandles.

We compute the third cohomology group of a dihedral quandle of odd order with finite field coefficient, generalizing our previous work. In particular, we obtain many new examples of non-trivial 3-cocycles. We check non-triviality of the associated invariant for knots.

For a knot K with a framing F, we show that the knot rack R(K, F) of (K, F) has the structure of a rack extension of the knot quandle Q(K), via the projection canonically obtained by disregarding its framing. From this fact, we can construct a cohomology class [ϕ_F] of Q(K) corresponding to the framing: a cohomology class in for K non-trivial, or one in for K trivial, where w_F denotes the writhe of (K, F). Moreover, when K is non-trivial, [ϕ_F] has a natural decomposition to the sum of and , where [ψ] is independent to F. This correspondence gives a bijection between the set of framings of K and .

This paper presents the first complete calculation of the cohomology of any nontrivial quandle, establishing that this cohomology exhibits a very rich and interesting algebraic structure. Rack and quandle cohomology have been applied in recent years to attack a number of problems in the theory of knots and their generalizations like virtual knots and higher-dimensional knots. An example of this is estimating the minimal number of triple points of surface knots [E. Hatakenaka, An estimate of the triple point numbers of surface knots by quandle cocycle invariants, Topology Appl139(1–3) (2004) 129–144.]. The theoretical importance of rack cohomology is exemplified by a theorem [R. Fenn, C. Rourke and B. Sanderson, James bundles and applications, Proc. London Math. Soc. (3)89(1) (2004) 217–240] identifying the homotopy groups of a rack space with a group of bordism classes of high-dimensional knots. There are also relations with other fields, like the study of solutions of the Yang–Baxter equations.

In this paper, from the viewpoint of quandle construction we prove that for any link type L and any diagram of it, there exists another diagram of L such that these two diagrams are Ω₁-dependent, Ω₂-dependent and Ω₃-dependent.