Questions tagged [quandles]
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18 questions
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Is there a Dehn-like presentation of a knot quandle?
The knot group can be presented using either a Wirtinger presentation (with generators corresponding to arcs of the knot diagram) or a Dehn presentation (with generators corresponding to regions of ...
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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: ...
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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 ...
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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 $K_1$ ...
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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 ...
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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.
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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_{\...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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) $(a*b) /b=(a/b)*b=a$
...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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