Questions tagged [knot-theory]

For questions on knot theory, the study of mathematical knots and their properties.

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, ${\Bbb R}^3$. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of ${\Bbb R}^3$ upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

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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 hopefully this question should be easier. There…
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What knot is this?

My headphone cables formed this knot: however I don't know much about knot theory and cannot tell what it is. In my opinion it isn't a figure-eight knot and certainly not a trefoil. Since it has $6$ crossings that doesn't leave many other…
s.harp
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What is knot theory about, exactly?

"In topology, knot theory is the study of mathematical knots." This is how Wikipedia defines knot theory. I have no idea what this is supposed to mean, but it does seem interesting. The rest of the article is full of examples of knots, their…
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Why are all knots trivial in 4D?

A classical knot is defined to be an embedding $S^1 \to \mathbb{R}^3$ where $S^1$ is a 1-sphere or circle. Embeddings $S^1 \to \mathbb{R}^4$ are usually not considered knots because they are trivial knots, i.e., they can be continuously deformed to…
Minethlos
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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 this works backwards, where I might have a really…
Kainui
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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.
Ritwik Bose
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Find $f$ such that $f^{-1}(\lbrace0\rbrace)$ is this knotted curve (M.W.Hirsh)

I would like to solve the following problem (it comes from Morris W. Hirsh, Differential Topology, it's exercise 6 section 4 chapter 1): Show that there is a $C^\infty$ map $f:D^3\to D^2$ with $0\in D^2$ as a regular value such that…
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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 $S^2 \to \mathbb R^4$ (differentiable)?
user58512
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Isotopy and Homotopy

What is the difference between homotopy and isotopy at the intuitive level.Some diagrammatic explanation will be helpful for me.
Koushik
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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 not, is there at least some similar algebraic curve…
Martin Brandenburg
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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 $\pi_3(S^2)=\mathbb{Z}$ can be understood as describing the…
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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 isotopic if there is an isotopy $h:\mathbb{R}^3…
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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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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 knows even less than I do about it, apparently.…
Shaun
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Why do we often consider knots to be embedded in $S^3$ instead of $\mathbb{R}^3$?

When doing knot theory toward the end of my algebraic topology course, we often defined knots as embeddings of $S^1$ in $S^3$ instead of $\mathbb{R}^3$. My professor justified this by saying that $S^3$ is just $\mathbb{R}^3$ with a point added at…
Osama Ghani
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