For questions on knot theory, the study of mathematical knots

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36
votes
1answer
384 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 ...
22
votes
3answers
462 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 ...
15
votes
0answers
156 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 ...
14
votes
1answer
181 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 ...
12
votes
2answers
361 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 ...
12
votes
4answers
812 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 ...
11
votes
1answer
448 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 ...
10
votes
8answers
977 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.
10
votes
1answer
317 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. ...
9
votes
2answers
276 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
119 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 ...
8
votes
2answers
258 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 $H^2$ ...
7
votes
4answers
264 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
384 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
2answers
399 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
0answers
175 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 ...
7
votes
2answers
297 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
181 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
118 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
2answers
332 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
301 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 ...
6
votes
1answer
72 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
1answer
159 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
2answers
149 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 ...
5
votes
1answer
80 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
154 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 ...
5
votes
0answers
279 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
3answers
598 views

School project in knot theory

Can someone suggest an idea for a school project in knot-theory for a 13 year old? Thanks
4
votes
1answer
143 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 ...
4
votes
2answers
206 views

A puzzle on knotted surfaces

Only after having learned that the somehow only notion of equivalence of knots is definitely "ambient isotopy" I stumbled over this blog entry on ambient isotopy. (Had it been earlier!) What bothers ...
4
votes
2answers
171 views

What is smoothness needed for?

We can either define a knot to be (1) a smooth embedding $S^1 \hookrightarrow \mathbb R^3$ or (2) a piecewise linear, simple closed curve in $\mathbb R^3$ Then these two definitions are ...
4
votes
2answers
250 views

Self-Linking Number on 3-Manifolds

We can assign a framing to a knot $K$ (in some nice enough space $M$) in order to calculate the self-linking number $lk(K,K)$. But of course it is not necessarily canonical, as added twists in your ...
4
votes
1answer
65 views

How to make a $C^1$ knot into a $C^\infty$ knot

Suppose I have a $C^1$ imbedding $f: S^1 \rightarrow S^3$. From the point of view of knot theory, what's the "best" way to get a $C^\infty$ curve that "looks like" or is "equivalent to" $f$? For ...
4
votes
2answers
98 views

Burau matrix of braid

What is the definition of a Burau matrix of a braid? Where can I find a definition?
4
votes
1answer
424 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 ...
4
votes
2answers
100 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 ...
4
votes
1answer
30 views

Is a knot shadow always compatible with the trivial knot?

Define a knot shadow as a projection of a knot that does not indicate over- and under-crossings. So, if there are $c$ crossings, there are $2^c$ possible over/under assignments, and so that many ...
4
votes
1answer
64 views

Alexander–Briggs notations for the links or knots of $N^3_m$

We can use Alexander–Briggs notations for the links or knots. For example, is three separate loops with no links. And there are many other examples of Alexander–Briggs notations for three ...
4
votes
2answers
83 views

Recommended books on knot invariants

I've been reading the books "An introduction to knot theory" by Lickorish and "Knots, Links, Braids and 3-Manifolds" by Prosolov and Sossinsky, and while both seem to me as good books, sometimes I'd ...
4
votes
1answer
239 views

Computing knot/link groups

The knot group of a knot $K$ is the fundamental group of $\mathbb R^3 \smallsetminus K$; that is, the set of possibly self-crossing closed paths (starting and ending at any single point in space) ...
4
votes
1answer
82 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 ...
4
votes
1answer
60 views

Computing the volume of a hyperbolic knot

Could anyone show me or refer me to a link where the volume of a hyperbolic knot, say, the figure-8 knot, is computed (well, in fact estimated) explicitly and not only having the procedures outlined?
4
votes
1answer
160 views

Assigning alternate crossings to closed curves

This is a minor curiosity that I've been wondering about. Suppose that we draw a closed curve in the plane and that this curve intersects itself several times, but never twice in one spot. We can knot ...
4
votes
1answer
117 views

What knot groups are Abelian?

The knot group (the fundamental group of the complement of a knot) of the unknot is $\mathbb{Z}$ and the Hopf link is $\mathbb{Z}^2$, so those are knots (links) with Abelian knot group but are there ...
4
votes
1answer
332 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. ...
4
votes
1answer
42 views

What does the tensor product in the definition of Combinatorial Floer knot homology look like?

I am working on a project that involves summarizing the article A combinatorial description of knot Floer homology (http://arxiv.org/abs/math/0607691) and doing some example computations with the ...
4
votes
1answer
110 views

Finding the Alexander polynomial of the following braid closure

How do you find the Alexander polynomial of the closure of the following braid, $(\sigma_1^{-2}\sigma_2^{-1}\sigma_3^{-1}\sigma_4^{-1}\sigma_5^{-1}...\sigma_{A-1}^{-1})^B$ where $A$ and $B$ are ...
4
votes
1answer
33 views

boundary map in the (M-V) sequence

Let $K\subset S^3$ be a knot, $N(K)$ be a tubular neighborhood of $K$ in $S^3$, $M_K$ to be the exterior of $K$ in $S^3$, i.e., $M_K=S^3-\text{interior of }{N(K)}$. Now, it is clear that $\partial ...
4
votes
1answer
59 views

Reidemeister III and minimal crossing knot

If you have a knot which has minimal crossings, can you do a Reidemeister III move? Thanks
4
votes
1answer
216 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) ...