All Questions
Tagged with quandles knot-theory
8 questions
2
votes
1
answer
136
views
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 ...
- 53.3k
7
votes
0
answers
362
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 ...
- 4,597
4
votes
2
answers
265
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 $K_1$ ...
- 4,597
9
votes
2
answers
873
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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 ...
- 63.9k
3
votes
1
answer
427
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 ...
- 39.9k
5
votes
1
answer
403
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 ...
- 63.9k
4
votes
1
answer
244
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 ...
- 63.9k
2
votes
1
answer
193
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 ...
- 1,108