Classifies spatial graphs with finite N-quandles.
problem Determining isomorphism of spatial graphs' N-quandles.
method Generalized N-quandles to spatial graphs, proving basic results and conjecturing a classification.
result Verifies conjecture in several cases, presents a possible counterexample.
Classifies links with finite N-quandles for some N.
problem Describing finite quotients of quandles, especially n-quandles. method Introducing and studying N-quandles, proving one direction of a conjectured classification. result Proves one direction of a conjectured classification of links with finite N-quandles. Complete classification of links and spatial graphs with finite N-quandles.
problem Classifying links and spatial graphs with finite N-quandles.
method Extending fundamental quandle relationships to N-quandles of links and spatial graphs.
result Complete list of links and partial list of spatial graphs with finite N-quandles.
This paper computes the second quandle homology group of knot n-quandles.
problem Characterizing knots using quandle homology groups.
method Computation of the second quandle homology group for knot n-quandles.
result The second quandle homology group of knot n-quandles provides more information than the knot quandle's homology group.
We compute Cayley graphs and automorphism groups for all finite n-quandles of two-bridge and torus knots and links, as well as torus links with an axis.
The fundamental n-quandles of links are residually finite for n ≥ 2.
problem Residual finiteness of fundamental n-quandles of oriented links.
method Investigation of residual finiteness and subquandle separability of quandles; use of Winker's work on 3-sphere branched covers.
result Fundamental n-quandles of oriented links are residually finite for each n ≥ 2.
We prove a conjecture of Przytycki which asserts that the n-quandle of a link L in the 3-sphere is finite if and only if the fundamental group of the n-fold cyclic branched cover of the 3-sphere, branched over L, is finite.
To better understand the fundamental quandle of a knot or link, it can be useful to look at finite quotients of the quandle. One such quotient is the n-quandle (or, when n=2, the {\em involutory} quandle). Hoste and Shanahan \cite{HS2} gave a complete list of the links which have finite n-quandles; it remained to…
Given a presentation for a rack R, we define a process which systematically enumerates the elements of R. The process is modeled on the systematic enumeration of cosets first given by Todd and Coxeter. This generalizes and improves the diagramming method for n-quandles introduced by Winker. We p…
Two quandles from Coxeter groups studied, showing similarities in automorphism groups.
problem Understanding quandles from Coxeter groups and their automorphism groups.
method Defined two families of quandles from Coxeter quandles and Pin groups, analyzed their automorphism groups.
result Similarity in inner automorphism groups of the two quandles.
The paper characterizes L-space 3-manifolds using quandle orderability.
problem Characterizing L-space 3-manifolds using quandle orderability.
method Using quandles and their extensions, the paper investigates orderability and circular orderability conditions.
result The n-quandle Qn(L) of a link quandle is not right circularly orderable. The paper studies finiteness of canonical quotients in Dehn quandles of surfaces.
problem Finiteness of canonical quotients in Dehn quandles of surfaces.
method Analyzing the Dehn quandle structure and using quandle quotients.
result For surfaces of positive genus, the n-quandle of their Dehn quandles is finite for specific values of n.