Questions tagged [quandles]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
10 votes
0 answers
180 views

Symmetric spaces are quandles. Is this important?

For concreteness, let's consider a connected reductive Lie group $G$, and an involution $\theta$ on it. Then the associated symmetric space $X=G/G^\theta$ has the structure of an involutive quandle: ...
7 votes
0 answers
318 views

When do two knots have isomorphic fundamental bikeis?

A kei, also known as an involutive (or involutory) quandle, is a quandle $(Q,*)$ satisfying the involution condition that $(x*y)*y=x$ for all $x$ and $y$. Just like we can define a fundamental ...
4 votes
2 answers
211 views

Can different knots have the same numbers of quandle colorings for all quandles?

Let $K_1$ and $K_2$ be two knots such that for all finite quandles $X$, the number of colorings of $K_1$ by $X$ is the same as the number of colorings of $K_2$ by $X$. Then my question is, must $K_1$ ...
3 votes
1 answer
300 views

Distinguishing Square Knot and Granny Knot using Quandles

It is known that the square knot and the granny knot are nonequivalent although they have isomorphic fundamental groups. I want to write a work on knot theory and provide these knots as an example ...
  • 133
7 votes
1 answer
260 views

Is the category of racks semi-abelian?

I wonder whether the category of (pointed) racks is semi-abelian. Any comments and references would be appreciated.
2 votes
0 answers
102 views

Is the action of free self-distributive algebras on racks computable in polynomial time?

Let $B_{\infty}$ denote the infinite strand braid group. Let $\mathrm{sh}:B_{\infty}\rightarrow B_{\infty}$ be the mapping where $\mathrm{sh}(\sigma_{i})=\sigma_{i+1}$ whenever $i\geq 1$. Then $B_{\...
12 votes
0 answers
242 views

Higher homotopical information in racks and quandles

A quandle is defined to be a set $Q$ with two binary operations $\star,\bar\star\colon\ Q\times Q\to Q$ for which the following axioms hold. Q1. a $\star$ a = a Q2. (a $\star$ b) $\bar\star$ b = (a $\...
4 votes
1 answer
191 views

Quandle homomorphism does not always induces group homomorphism on inner automorphism groups of quandles

Let $X$ and $Y$ be two quandles and $f: X \rightarrow Y$ be a quandle homomorphism. Then we can define a map $\bar f: Inn(X) \rightarrow Inn(Y)$ as $\bar f(S_a)=S_{f(a)}$, where $a \in X$. Then $\bar ...
  • 163
2 votes
1 answer
184 views

Classification of pretzel links up to link homotopy using alexander quandle

I am currently reading this paper where the author classifies the pretzel links up to link homotopy using a quasi-trivial quandle $\mathbb{Z}_{k}[t^{\pm 1}]\diagup_{(t-1)^{2}}$, and I find it ...
  • 55
3 votes
0 answers
208 views

Partial information decomposition for tangle machines

In (Williams and Beer, 2010), they define the partial information decomposition (PID) as a generalization of Shannon's Mutual Information for multiple information sources. Their key insight is that ...
  • 211
7 votes
0 answers
308 views

Hemi-semi direct product of racks or quandles

In the category of racks (similarly quandles), instead of well-known semidirect product, we have the hemi-semi direct product construction as seen on Wagemann & Crans. As far as I know, semi ...
10 votes
4 answers
1k views

Conjugation Quandles and... "Quandle-Groups"? From quandles to Groups

This question is already asked MathSE A quandle $(Q,*,/ )$ is a idempotent right-distributive and right invertible structure. 1) $a*a=a$ 2) $(a*b)*c=(a*c)*(b*c)$ 3) $(a*b) /b=(a/b)*b=a$ ...
  • 233
3 votes
2 answers
635 views

A name for the inverse image of the center of a quotient group?

Given the projection $\pi_A$ from a group $G$ to $G/A$ where $A$ is normal, is there a name and/or a standard notation for $\pi_A^{-1}\left(Z\left(G/A\right)\right)$? I came across this object in my ...
5 votes
1 answer
375 views

One question about the quandle

Given a finite quandle $Q$, for any knot $K$ one can associate an invariant, i.e. the number of proper colorings $p(K)$. Let us consider the inverse $K^{-1}$ and mirror image $K'$ of $K$. My queston ...
8 votes
1 answer
321 views

The equality problem between conjugate group elements

The Novikov--Boone Theorem, which is perhaps the archetypal local unsolvability result in group theory, states existence of a finitely presented group whose word problem is recursively unsolvable. ...
15 votes
2 answers
566 views

Formally undecidable problems on finitely presented quandles

In the literature, one sometimes sees the claim that finitely presented quandles (in particular, knot quandles) are "hard to deal with". Hence, a great deal of effort has gone into studying finite ...
  • 1,837
8 votes
2 answers
809 views

Higher order quandle

The notion of quandle is known to be closely related to knot theory. The three axioms in the definition of quandle correspond to the Reidemeister moves. Recently I learned that there are higher ...
  • 1,417