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Questions tagged [quandles]

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10
votes
0answers
136 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
1answer
99 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 ...
0
votes
0answers
55 views

show that the set of functions from $ \mathbb{Z}[X^{n}] $ to $ A $ is generated by the characteristic functions

Let $ X $ be a quandle and $ A $ be an abelian group (for simplicity assume that $ A $ is a finite cyclic group $ \mathbb{Z}_{n} $ or the infinite cyclic group $ \mathbb{Z} $). I need to show that the ...
2
votes
1answer
125 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 ...
3
votes
0answers
162 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 ...
1
vote
0answers
40 views

Examples of partially permutative left-distributive algebras

An algebra $(X,*)$ that satisfies the identity $x*(y*z)=(x*y)*(x*z)$ is said to be a left-distributive algebra. Let $L:X^{2}\rightarrow X^{2}$ be the mapping defined by $L(x,y)=(x*y,x)$ and let $T:X^{...
7
votes
4answers
843 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) $(a*b) /b=(a/b)*b=a$ ...
3
votes
2answers
501 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 ...
4
votes
1answer
260 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
1answer
300 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. ...
14
votes
2answers
470 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 ...