13
votes
Is the category of racks semi-abelian?
The category $\mathbf{Rack}$ of racks is Barr-exact since it is a variety of universal algebras, but it is not protomodular. Indeed, the category of sets is equivalent to the category of racks ...
- 555
11
votes
Accepted
Distinguishing Square Knot and Granny Knot using Quandles
See Example 3.6 of Ellis and Fragnaud, Computing with knot quandles. It says,
The following commands use the quandle invariant ColQ(K) to establish that the granny knot is not equivalent to the ...
- 39.9k
5
votes
Accepted
Can different knots have the same numbers of quandle colorings for all quandles?
The short answer is that I think this is an open question.
This is stated as Conjecture 3.4 here, proved for knots up to 12 crossings:
Clark, W. Edwin; Elhamdadi, Mohamed; Saito, Masahico; Yeatman, ...
- 68.9k
3
votes
Accepted
Quandle homomorphism does not always induces group homomorphism on inner automorphism groups of quandles
Let's make sure we agree on definitions.
A quandle is an algebraic structure $(A,*)$ for which
each right multiplication map $S_a(x)= x*a$ by an element $a\in A$
is an automorphism fixing $a$. These ...
- 14.6k
2
votes
Accepted
Classification of pretzel links up to link homotopy using alexander quandle
Beginning with coloring the $k$th box using
${\mathbb{Z}_k[t^{\pm1}]}/_{(t-1)^2}$, we conclude that if the link is
homotopically trivial, then $p_k$ divides all $p_i$'s.
It seems that there is a ...
- 1,108
1
vote
Accepted
Is there a Dehn-like presentation of a knot quandle?
Reposting an answer from a deleted account:
Yes, there is an analogous Dehn-like presentation for knot quandles where generators correspond to regions of a knot diagram instead of arcs. This ...
1
vote
Can different knots have the same numbers of quandle colorings for all quandles?
Not a direct answer but a related one:
The granny knot and the square knot are known to have isomorphic $\pi_1$, but they can be detected by quandle coloring.
An application of that is the ...
- 1,028
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