For questions on knot theory, the study of mathematical knots

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36
votes
1answer
456 views

Can I solve an integral (or other tough problem) by playing with knots?

I've seen that in calculating things in knot theory that involves a lot of hard looking integrals and matrices, even though the knots themselves appear fairly simple. So is there some way in which ...
23
votes
3answers
562 views

Picture of a 4D knot

A knot is a way to put a circle into 3-space $S^1 \to \mathbb R^3$ and these are often visualized as 2D knot diagrams. Can anyone show me a diagram of a nontrivial knotted sphere $S^2 \to \mathbb ...
19
votes
0answers
242 views

Geometric way to view the truncated braid groups?

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question. I also asked a related question on MO, although ...
15
votes
1answer
197 views

Visualize Fourth Homotopy Group of $S^2$

I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in ...
14
votes
9answers
2k 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.
13
votes
5answers
1k views

Trefoil knot as an algebraic curve

Is the trefoil knot with its usual embedding into affine $3$-space an algebraic curve (maybe after extending scalars to $\mathbb{C}$)? Is there even some thickening to some algebraic surface? If ...
12
votes
2answers
404 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 ...
11
votes
1answer
732 views

Why is the knot group of the trefoil isomorphic to the group of 3-braids?

I apologise in advance for the vagueness of this question but I have not been able to find very much info on the topic and have made very little progress on my own. I am trying to understand why the ...
11
votes
1answer
352 views

Can the n-string sphere braid group embed in to the (n+1)-string sphere braid group?

This question has been cross posted on MathOverflow with some very interesting answers and discussion. I'm currently writing a project on the braid groups and their analogues on closed surfaces. ...
11
votes
0answers
182 views

Why is the Mazur swindle named so?

Often results or techniques in mathematics are called 'theorems'. Sometimes they are called 'tricks'. In no other context have I seen a result called a 'swindle'. Is there a historical reason for this ...
10
votes
5answers
422 views

An Illustrated Classification of Knots.

Let me be honest here: I know very little about Knot Theory. I'm sorry. I've a friend though, someone with no training in Mathematics at all but who is a huge fan of knots (for whatever reason), who ...
9
votes
3answers
435 views

How to start learning knot theory?

Knot theory really sounds cool and I'm very interested in it. But I'm wondering what basic knowledge it is required and how I should start learning about it. Thanks
9
votes
0answers
209 views

How did Chern pictured the first Chern number?

The first Chern number $\cal C$ is known to be related to various physical objects. Gauge fields are known as connections of some principle bundles. In particular, principle $U(1)$ bundle is said to ...
8
votes
1answer
708 views

Braid groups and the fundamental group of the configuration space of $n$ points

I am giving a lecture on Braid Groups this month at a seminar and I am confused about how to understand the fundamental group of the configuration space of $n$ points, so I will define some ...
8
votes
2answers
161 views

Jones Polynomial from Statistical Mechanics

I've been told that, given a knot projection, there is a way of associating a statistical system in such a way that the partition function of the system corresponds to the Jones polynomial of the ...
8
votes
2answers
320 views

Total mean curvature of an immersed torus.

How to prove that the total mean curvature of an immersed torus of $R^3$ such that has nontrivial self-intersection must $> 8 \pi$? The definition of total mean curvature is the integral of $H^2$ ...
8
votes
2answers
570 views

Seifert matrices — Figure 8 knot

I've just learnt about Seifert matrices and thought it might be a good idea to compute some. Can you tell me if this is right: Here $x_1^+$ denotes the push off of $x_1$. I have omitted the diagram ...
7
votes
2answers
233 views

Which mathematical topics is knot theory related to?

I wonder if knot theory is related to any other topic in mathematics. I've not read much about it, but it seems to be living isolated. I also wonder if there any particular mathematical background ...
7
votes
1answer
135 views

Equivalence of knots: ambient isotopy vs. homeomorphism

I am looking into knot theory and have found two different definitions stating that two knots $K_1$ and $K_2$ are equivalent, namely the concept of an ambient isotopy: These two knots are ambient ...
7
votes
3answers
704 views

An introduction to Khovanov homology, Heegaard-Floer homology

I am interested in knot theory and low dimensional topology. I would like to start studying Khovanov homology and Heegaard-Floer homology. I (partially) read the original paper of Khovanov and then ...
7
votes
4answers
312 views

Reference for an unknotting move

Consider the following move on diagrams. I dimly recall hearing or reading that a sequence of such moves is sufficient to unknot any knot but I don't recall where I saw this. The strands in the ...
7
votes
1answer
521 views

Complement of figure-8 knot

I am reading W. Thurston's famous "3-dimensional Geometry and Topology", but I am stuck at the point where it is said that gluing two tetrahedra in an appropriate way give you the complement of the ...
7
votes
0answers
68 views

Is “slightly deform” a well defined concept in mathematical proof?

In topological proofs the phrase "slightly deform" is widely used. To me, although I can accept the idea intuitively, the phrase "slightly deform" does not sound like a strict mathematical concept. ...
7
votes
2answers
354 views

Seifert matrices and Arf invariant — Cinquefoil knot

I have computed the following Seifert matrix for the Cinquefoil knot: $$ S = \begin{pmatrix} 1 & -1 & -1 & -1 \\ 0 & 1 & 0 & 0 \\  0 & 1 & 1 & 0 \\ 0 ...
6
votes
2answers
195 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 ...
6
votes
2answers
190 views

Framed Cobordism Classes of links in $\mathbb R^3$

We know that every link in $S^3$ is framed cobordant to the unknot with some framing. The idea is to study smooth homotopy classes of maps from $S^3$ to $S^2$. Actually in the title I have given ...
6
votes
2answers
123 views

How does smoothness prevent “singularities”?

This is a refinement of one of my earlier questions (I failed to put into words what I really wanted to ask). First of all, I'm not sure "singularity" is the correct word to use hence the quotes. ...
6
votes
1answer
177 views

How to ensure Topological Correctness

Question: I read through an enormous amount of material on topology and knot-theory in wikipedia, but I still am stuck at the following fundamental problem: Given two representations of closed ...
6
votes
2answers
369 views

Uniqueness of Preferred Framing of a Solid Torus in $S^3$

One way to state my question tersely is: For a homeomorphism $f : S^1 \times \mathbb{D}^2 \rightarrow S^1 \times \mathbb{D}^2$, does $f|_{S^1 \times S^1}$ determine the isotopy class of $f$? This is ...
6
votes
1answer
89 views

Probability that two circles in space are linked

Let $C_0$ be a circle centered on the origin, and $C_1$ a circle centered on $(1,0,0)$, center distance of $1$. Q1. If both $C_0$ and $C_1$ are randomly oriented and have the same radius $r ...
6
votes
2answers
209 views

Knots and graphs

Every knot gives rise to a number of 4-regular planar graphs - by regular projections onto the plane - which just have to be enriched by an over/under flag for every vertex to be able to reconstruct ...
5
votes
3answers
428 views

Framing Integer associated with a Framed Knot/Link

I have been looking for a clear definition of the n-framing of a knot unsuccessfully. A framing here refers to a choice of homeomorphism between a solid torus neighborhood (a.k.a, tubular ...
5
votes
3answers
684 views

School project in knot theory

Can someone suggest an idea for a school project in knot-theory for a 13 year old? Thanks
5
votes
1answer
535 views

Framed manifolds and framed knots

I've been looking at intersection forms and the Arf invariant recently and I got a comment to one of my previous questions related to this. So I looked at framed manifolds. There seems to be quite a ...
5
votes
1answer
164 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 ...
5
votes
1answer
92 views

Can a trefoil knot be stretched to look like a triangle with three knots at the vertices?

Can a trefoil knot be stretched to look like a triangle with three knots at the vertices, like in the right side of the image below, or is that transformation impossible to happen? If possible, what ...
5
votes
1answer
73 views

How to prove a knot with genus larger than 1 is prime, such as Miller Institute Knot?

It is easy to show that a knot with genus 1 is a prime knot because the genus is additive under direct sum. However, I found that some prime knot, for example, $6_2$ the Miller Institute Knot have ...
5
votes
2answers
297 views

Homology of knot complement

I was told in a topology class that if $Y$ is a closed $3$-manifold and $K$ is a null-homologous knot in $Y$, then $H_1(Y- \nu(K)) \cong H_1(Y) \oplus \mathbb{Z}$. I'm trying to prove this ...
5
votes
2answers
197 views

Obtaining Wirtinger presentation using van Kampen theorem

Hatcher's Algebraic Topology, section 1.2, problem #22 describes an algorithm for computing the Wirtinger presentation of the complement of a smooth or piecewise linear knot K in $\mathbb{R}^3$: ...
5
votes
1answer
200 views

Category theory for knot theory

Where would I start to learn category theory for its use in knot theory? I have a background in physics and Ive read Adams Knot book. I know nothing about category theory. Eventually I want to learn ...
5
votes
1answer
126 views

Showing every knot has a regular projection using diff top

My question is: Can we use differential topology to prove that every smooth knot has a regular projection? Here is some background: Let $\gamma : S^1 \rightarrow \mathbb{R}^3$ be a smooth unit-speed ...
5
votes
1answer
287 views

Question about definition of regular projection of a knot

One can define a knot in two ways: (1) A knot is a closed polygonal curve in $\mathbb R^3$ (2) A knot is an equivalence class of embeddings $S^1 \hookrightarrow \mathbb R^3$ And perhaps also: (3) ...
5
votes
0answers
79 views

Twisted nail puzzle framed in terms of algebraic topology?

See here for a description of the puzzle: Twisted Nail Puzzle. My question is, can someone provide a description of the puzzle and its solution in context of the language of algebraic topology?
5
votes
0answers
50 views

Can information about a knot be recovered from the Jones Polynomial?

Suppose we know the Jones polynomial of some knot, but maybe not specifically which knot. Can any information about the knot be recovered just by knowing its Jones polynomial? Say, for example, the ...
5
votes
0answers
357 views

Definition of Reshetikhin-Turaev TQFT

I am studying Reshetikhin-Turaev TQFT. In their paper or in the book " Quantum invariants of knots and 3-manifolds", they first define an invariant $\tau(M)$ for a closed orientable 3-manifold $M$ and ...
4
votes
1answer
54 views

Is there a one to one correspondence between Jones' polynomials and knots?

I know Jones' polynomial is a knot invariant. By using knot invariant like p-coloration one can only say whether two knots are different but not whether they are the same. So it is like injective ...
4
votes
2answers
185 views

homeomorphism of cantor set extends to the plane?

Suppose C is a Cantor set in the Euclidean plane, or even in R^3. Suppose h is a homeomorphism of C onto itself. Can h be extended to a homeomorphism of the whole space? What about if h preserves the ...
4
votes
1answer
646 views

Isotopy and Homotopy

What is the difference between homotopy and isotopy at the intuitive level.Some diagrammatic explanation will be helpful for me.
4
votes
1answer
372 views

Relation between the braid group and the mapping class group of the plane

According to the following link, page 248, the braid group modulo its center is isomorphic to the mapping class group of the $N$-times punctured plane, i.e. $B_N/Z(B_N)\cong M_N(\mathcal(R)^2)$. Could ...
4
votes
1answer
40 views

Ways to link the unknot to a pole

Is there a way to show that the following ways of linking an unknot to an infinite horizontal pole are inequivalent? Perhaps the Wirtinger presentation would work, but I am not sure because of the ...