For questions on knot theory, the study of mathematical knots

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2
votes
1answer
150 views

Locally flat submanifold

Recently I found the following definition: Let $M^{n}$ be an $n-$dimensional topological manifold. Then $N^{k}\subseteq M^{n}$ is a locally flat submanifold if for every $x\in N$ there exists an open ...
4
votes
1answer
139 views

knot invariants from the symmetric group

Producing knot/link invariants is "as simple as" finding functions on the braid group invariant under the Markov moves. Many classical invariants arise from the character of a representation of the ...
1
vote
1answer
128 views

Are satellite knots prime?

Which satellite knots are prime? I do know that connected sum of knots is a satellite operation, but I found this statement: "the satellite knots all have structures which are well known and ...
4
votes
0answers
93 views

Knots with Fox tricoloring number tri(K)=27

I would be very grateful if you help me to find such knots. Or to find a knot atlas, where this invariant is included. I tried to find, but did not sucseed(
0
votes
1answer
64 views

Positive genus and Abelian-ness

So if a Seifert surface of a knot K has positive genus, what can we say about the its complement's first fundamental group, is it Abelian because the only knot with Seifert surface of positive knot is ...
0
votes
3answers
394 views

restrictions on the fundamental group of a knot complement?

Are there any restrictions on $\pi_1(S^3\backslash K)$ for a (tame) knot $K$ besides having $\pi_1^{\text{ab}}(S^3\backslash K)=\mathbb{Z}$? So we have 1) finite presentation, 2) $H_1=\mathbb{Z}$, 3) ...
4
votes
1answer
328 views

Pretzel knot equivalence

How would you go about proving the $(p, q, r)$ -pretzel knot is equivalent to the $(p, r, q)$ -pretzel knot? By "equivalent" I mean you can change one knot into the other by elementary deformations. ...
12
votes
2answers
345 views

Why are knot invariants best organized as polynomials?

Does anyone have a good explanation for why Knot invariants tend to be well organized as polynomials? What exactly is going on and why don't we often see polynomial invariants for classifying other ...
6
votes
2answers
172 views

The construction of knotted surfaces in $\mathbb{R}^4$

For a two-sphere embedded in $\mathbb{R}^4$,how can you check whether or not there is an ambient isotopy to the "standard" 2-sphere (the set of points $(x,y,z,0)$ in $\mathbb{R}^4$ distance 1 from the ...
3
votes
3answers
575 views

School project in knot theory

Can someone suggest an idea for a school project in knot-theory for a 13 year old? Thanks
10
votes
8answers
856 views

A good quick introduction to Knot Theory?

Is there a good quick introduction to knot theory? I am relatively mathematically savvy so any level is appreciated.