K. Cho and S. Nelson introduced the notion of a quandle coloring quiver, which is a quiver-valued link invariant, and a quandle cocycle quiver which is an enhancement of the quandle coloring quiver by assigning to each vertex a weight computed using a quandle 2-cocycle. In this paper, we study quandle coloring quivers using dihedral quandles and introduce the notions of a shadow quandle coloring quiver and a shadow quandle cocycle quiver, which are shadow versions of the quandle coloring quiver and the quandle cocycle quiver. We show that, when we use a dihedral quandle of prime order, the quandle coloring quivers are equivalent to the quandle coloring numbers and when we use a dihedral quandle of prime order and Mochizuki's 3-cocycle, the shadow quandle cocycle quivers are equivalent to the shadow quandle cocycle invariants.
A quandle coloring quiver is a quiver structure, introduced by Karina Cho and Sam Nelson, which is defined on the set of quandle colorings of an oriented knot or link by a finite quandle. We study quandle coloring quivers of (p, 2)-torus knots and links with respect to dihedral quandles.
We incorporate quandle cocycle information into the quandle coloring quivers we defined in arXiv:1807.10465 to define weighted directed graph-valued invariants of oriented links we call \textit{quandle cocycle quivers}. This construction turns the quandle cocycle invariant into a small category, yielding a categorification of the quandle cocycleinvariant. From these graphs we define several new link invariants including a 2-variable polynomial which specializes to the usual quandle cocycle invariant. Examples and computations are provided.
We solve the conjecture by R. Fenn, C. Rourke and B. Sanderson that the rack homology of dihedral quandles satisfies H_3^R(R_p) = Z \oplus Z_p for p odd prime. We also show that H_n^R(R_p) contains Z_p for n>2. Furthermore, we show that the torsion of H_n^R(R_3) is annihilated by 3. We also prove that the quandle homology H_4^Q(R_p) contains Z_p for p odd prime. We conjecture that for n>1 quandle homology satisfies: H_n^Q(R_p) = Z_p^{f_n}, where f_n are "delayed" Fibonacci numbers, that is, f_n = f_{n-1} + f_{n-3} and f(1)=f(2)=0, f(3)=1. Our paper is the first step in approaching this conject…
Quandles with involutions that satisfy certain conditions, called good involutions, can be used to color non-orientable surface-knots. We use subgroups of signed permutation matrices to construct non-trivial good involutions on extensions of odd order dihedral quandles. For the smallest example of order 6 that is an extension of the three-element dihedral quandle, various symmetric quandle homology groups are computed, and applications to the minimal triple point number of surface-knots are given.
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 enhance the quandle coloring quiver invariant of oriented knots and links with quandle modules. This results in a two-variable polynomial invariant with specializes to the previous quandle module polynomial invariant as well as to the quandle counting invariant. We provide example computations to show that the enhancement is proper in the sense that it distinguishes knots and links with the same quandle module polynomial.
We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most literal sense, i.e., giving the set of quandle colorings the structure of a small category which is unchanged by Reidemeister moves. We derive some new enhancements of the counting invariant from this quiver structure and show that the enhancements are proper with explicit examples.
We enhance the psyquandle counting invariant for singular knots and pseudoknots using quivers analogously to quandle coloring quivers. This enables us to extend the in-degree polynomial invariants from quandle coloring quiver theory to the case of singular knots and pseudoknots. As a side effect we obtain biquandle coloring quivers and in-degree polynomial invariants for classical and virtual knots and links.
We give a construction of quandle cocycles from group cocycles, especially, for any integer p \geq 3, quandle cocycles of the dihedral quandle R_p from group cocycles of the cyclic group Z/p. We will show that a group 3-cocycle of Z/p gives rise to a non-trivial quandle 3-cocycle of R_p. When p is an odd prime, since dim_{F_p} H_Q^3(R_p; F_p) = 1, our 3-cocycle is a constant multiple of the Mochizuki 3-cocycle up to coboundary. Dually, we construct a group cycle represented by a cyclic branched covering branched along a knot K from the quandle cycle associated with a colored diagram of K.
We show that the lower bounds for Betti numbers given in math.GT/9909161 are equalities for a class of racks that includes dihedral and Alexander racks. We confirm a conjecture from the same paper by defining a splitting for the short exact sequence of quandle chain complexes. We define isomorphisms between Alexander racks of certain forms, and we also list the second and third homology groups of some dihedral and Alexander quandles.
