# Tagged Questions

For questions on knot theory, the study of mathematical knots

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### 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 ...
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### 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 ´...
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### 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 &...
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### Knot with genus $1$ and trivial Alexander polynomial?

I would like to know whether there exists a knot $K$ with genus $g(K)=1$ and trivial Alexander polynomial $\Delta(K) \doteq 1$. A linked question could be: does there exist a Whitehead double with ...
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### Laymans explanation of the relation between QFT and knot theory

Could someone give an laymans explanation of the relation between QFT and knot theory? What are the central ideas in Wittens work on the Jones polynomial?
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### 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 ...
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### 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 $S^2 \to \mathbb R^4$...
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### 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 ...
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### 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
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### 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 ...
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### 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 ...
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### Surgery to unlink $S^1$ and $S^2$ in $S^4$ [closed]

Let us start with a $S^1$ and a $S^2$ are linked in $S^4$. Can I unlink the $S^1$ and $S^2$ by doing some surgery (with certain constraints described below, and let us say both $S^1$ and $S^2$ ...
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### 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 ...
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### Homology of knot exterior in general manifold for not null-homologus knot

I was trying to figure out the homology of the knot complement when $K$ is not a rational null-homologous knot ($[K]\neq 0\in H_1(X_K,\mathbb Q)$). We then know by half-lives half dies that the ...
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### Are there Kirby diagrams for manifolds with boundaries?

There are Kirby diagrams for 3- and 4-manifolds which consist of framed links corresponding to 1- and 2-handles attached to a single 0-handle. Any such diagram will give a unique closed manifold since ...
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### Why is the Hopf link the only link with knot group $\mathbb{Z} \oplus \mathbb{Z}$?

We can use the Loop Theorem to show that if $\Sigma$ is a minimal-genus Seifert surface for a link $L$, then $\pi_1(\Sigma)$ injects into the knot group $\pi_1(S^3 \setminus L)$. An orientable ...
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### Equivalence of knots

It's intuitively clear what it means that two knots $K,K'$ are essentially the same, but it can be termed and defined more precisely in different ways. Are all of them equivalent? $K, K'$ are ...
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### A generalization of the connected sum of links

A connected sum of two links $K$ and $L$ involves cutting a segment in each link and joining them up as illustrated in the top diagram, the connected sum of two trefoil knots. Is there any terminology ...
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### How is PL knot theory related to smooth knot theory?

I really want to like knot theory but the PL condition seems sort of ugly. I was hoping someone could give me a justification for secretly thinking about smooth knots as I read through a book like ...
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### The Abelianization of $\langle x, a \mid a^2x=xa\rangle$

I wish to verify the following statement (which comes from Fox, "A Quick Trip Through Knot Theory", although that is probably not important). "$\Gamma=\pi_1 (M)=\langle x, a \mid a^2x=xa\rangle$ so ...
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### Braid invariants resource

What are some braid invariants (analogous to the idea of knot invariants) or a resource where I can find them?
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### Has knot theory led to the development of better knots?

Knot theory was likely originally motivated by the study of real-world knots such as these: Indeed, mathematical knot tables to this day look not too dissimilar from the familiar "age of sail"-...
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### 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 ...
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### The isotopy type of the $Pretzel$ three strand knot $P(a,b,c)$

I'm trying to prove that the knot $8_{20}$ is quasi-alternating. I thought of using the fact that this knot is actualy aquivalent to the $Pretzel$ knot $P(3,-3,2)$. There's this famous theorem that ...
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### Why is the Whitehead double of a knot always prime?

I was looking for a proof that there are infinitely many prime knots and one said "take your favorite (prime) knot and consider all its Whitehead double", implying that all Whitehead doubles of a (...
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### Prove that an infinitely long rope can only form slipknots

I've heard that an infinitely long rope can only form slipknots, is that true, and is there a simple proof/obvious counterexample? Answers requiring no preliminary knowledge about topology would be ...
Is the set of all piecewise linear (PL) knots is a good approximation of the set of all 1D smooth knots embedded in $\mathbb{R}^3$? Once I saw a theorem related to that but not able to find it now. ...