In CJKLS quandle cohomology is used to produce invariants for particular embeddings of codimension two; 2-cocycles give to invariants for (classical) knots and 3-cocycles give rise to invariants for knotted surfaces. This is done by way of a notion of coloring of a diagram. Also, these invariants have the form of state-sums (or partition functions) used in Statistical Mechanics.
By a careful analysis of these colorings of diagrams we are able to come up with new invariants which correspond to calculation of the partition function at finite temperatures.
We define a two-variable polynomial invariant of finite quandles. In many cases this invariant completely determines the algebraic structure of the quandle up to isomorphism. We use this polynomial to define a family of link invariants which generalize the quandle counting invariant.
We study the structure of symplectic quandles, quandles which are also R-modules equipped with an antisymmetric bilinear form. We show that every finite dimensional symplectic quandle over a finite field F or arbitrary field F of characteristic other than 2 is a disjoint union of a trivial quandle and a connected quandle. We use the module structure of a symplectic quandle over a finite ring to refine and strengthen the quandle counting invariant.
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 prescribe inner automorphism group. The latter theorem suggests a new approach to the classification of fini…
Quandle coloring quivers are directed graph-valued invariants of oriented knots and links, defined using a choice of finite quandle $X$ and set $S\subset\mathrm{Hom}(X,X)$ of endomorphisms. From a quandle coloring quiver, a polynomial knot invariant known as the \textit{in-degree quiver polynomial} is defined. We consider quandle coloring quiver invariants for oriented surface-links, represented by marked graph diagrams. We provide example computations for all oriented surface-links with ch-index up to 10 for choices of quandles and endomorphisms.
We define a family of generalizations of the two-variable quandle polynomial. These polynomial invariants generalize in a natural way to eight-variable polynomial invariants of finite biquandles. We use these polynomials to define a family of link invariants which further generalize the quandle counting invariant.
The state-sum invariants for knots and knotted surfaces defined from quandle cocycles are described using the Kronecker product between cycles represented by colored knot diagrams and a cocycle of a finite quandle used to color the diagram. Such an interpretation is applied to evaluating the invariants. Algebraic interpretations of quandle cocycles as deformations of extensions are also given. The proofs rely on colored knot diagrams.
We define ambient isotopy invariants of oriented knots and links using the counting invariants of framed links defined by finite racks. These invariants reduce to the usual quandle counting invariant when the rack in question is a quandle. We are able to further enhance these counting invariants with 2-cocycles from the coloring rack's second rack cohomology satisfying a new degeneracy condition which reduces to the usual case for quandles.
We define enhancements of the quandle counting invariant for knots and links with a finite labeling quandle Q embedded in the quandle of units of a Lie algebra \mathfrak{a} using Lie ideals. We provide examples demonstrating that the enhancement is stronger than the associated unenhanced counting invariant.
Carter, Jelsovsky, Kamada, Langford and Saito have defined an invariant of classical links associated to each element of the second cohomology of a finite quandle. We study these invariants for Alexander quandles of the form Z[t,t^{-1}]/(p, t^2 + kappa t + 1), where p is a prime number and t^2 + kappa t + 1 is irreducible modulo p. For each such quandle, there is an invariant with values in the group ring Z[C_p] of a cyclic group of order p. We shall show that the values of this invariant all have the form Gamma_p^r p^{2s} for a fixed element Gamma_p of Z[C_p] and integers r >= 0 and s > 0. We…
A birack is an algebraic structure with axioms encoding the blackboard-framed Reidemeister moves, incorporating quandles, racks, strong biquandles and semiquandles as special cases. In this paper we extend the counting invariant for finite racks to the case of finite biracks. We introduce a family of biracks generalizing Alexander quandles, $(t,s)$-racks, Alexander biquandles and Silver-Williams switches, known as $(τ,σ,ρ)$-biracks. We consider enhancements of the counting invariant using writhe vectors, image subbiracks, and birack polynomials.
We introduce a notion of ternary distributive algebraic structure, give examples, and relate it to the notion of a quandle. Classification is given for low order structures of this type. Constructions of such structures from ternary bialgebras are provided. We also describe ternary distributive algebraic structures coming from groups and give examples from vector spaces whose bases are elements of a finite ternary distributive set. We introduce a cohomology theory that is analogous to Hochschild cohomology and relate it to a formal deformation theory of these structures.
We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks, quandles, rack and quandle cocycles, 2-crossed modules and braided crossed modules. We prove that our construction includes all rack and quandle cohomology (framed) link invariants, as well as the Eisermann invariant of knots, for which we also find a lifting by using braided crossed modules.
We use the inflation-restriction sequence and a result of Etingof and Graña on the rack cohomology to give a explicit description of 2-cocycles of finite indecomposable quandles with values in an abelian group. Several applications are given.
We define invariants of unoriented knots and links by enhancing the integral kei counting invariant Phi_X^Z (K) for a finite kei X using representations of the kei algebra, Z_K[X], a quotient of the quandle algebra Z[X] defined by Andruskiewitsch and Grana. We give an example that demonstrates that the enhanced invariant is stronger than the unenhanced kei counting invariant. As an application, we use a quandle module over the Takasaki kei on Z_3 which is not a Z_K[X]-module to detect the non-invertibility of a virtual knot.
We introduce the notion of an `inverse property' (IP) quandle C which we propose as the right notion of `Lie algebra' in the category of sets. To any IP quandle we construct an associated group G_C. For a class of IP quandles which we call `locally skew' and when G_C is finite we show that the noncommutative de Rham cohomology H^1(G_C) is trivial aside from a single generator θthat has no classical analogue. If we start with a group G then any subset C\subseteq G\setminus {e} which is ad-stable and inversion-stable naturally has the structure of an IP quandle. If C also generates G then we sho…
This paper is motivated by a general question: for which values of k and n is the universal Burnside kei of k generators and Kei "exponent" n, $\bar Q(k,n)$, finite? It is known (starting from the work of M. Takasaki (1942)) that $\bar Q(2,n)$ is isomorphic to the dihedral quandle Z_n and $\bar Q(3,3)$ is isomorphic to Z_3 + Z_3. In this paper we give descriptions of $\bar Q(4,3)$ and $\bar Q(3,4)$. We also investigate some properties of arbitrary quandles satisfying the universal Burnside relation (of "exponent" n) a=...a*b*...*a*b. In particular, we prove that the order of a finite commutati…
The aim of this paper is to define a homology theory for racks with finite rank N and use it to define invariants of knots generalizing the CJKLS 2-cocycle invariants related to the invariants defined in [15]. For this purpose, we prove that N -degenerate chains form a sub-complex of the classical complex defining rack homology. If a rack has rack rank N = 1 then it is a quandle and our homology theory coincides with the CKJLS homology theory [6]. Nontrivial cocycles are used to define invariants of knots and examples of calculations for classical knots with up to 8 crossings and classical lin…
We introduce and study so-called self-indexed graphs. These are (oriented) finite graphs endowed with a map from the set of edges to the set of vertices. Such graphs naturally arise from classical knot and link diagrams. In fact, the graphs resulting from link diagrams have an additional structure, an integral flow. We call a self-indexed graph with integral flow a comte. The analogy with links allows us to define transformations of comtes generalizing the Reidemeister moves on link diagrams. We show that many invariants of links can be generalized to comtes, most notably the linking number, t…
