All Questions
Tagged with knot-theory quandles
8 questions
9
votes
2
answers
873
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 ...
- 1,467
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
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 ...
- 517
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
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 ...
- 163
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 ...
- 133
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 ...
- 55
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 ...
