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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.

1214 questions
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Ambient isotopy and isotopy

In the lecture note of my knot theory course, there is a very short explanation why we need ambient isotopy instead of just isotopy. It uses the following figure to illustrate without any explanation. I don't understand why it's OK to make a knot…
Dasheng Wang
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Which knot is this?

I am trying to figure out which knot this is. Is it a prime knot or a composite knot? If I am counting correctly, there are 8 crossings in this image. Is this knot 8_16 shown here? edit: I am starting to think this is 8_18. P.S. This the cover…
iamnsi
  • 81
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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 for $x_2$ since I think that it's the same as the…
Rudy the Reindeer
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Seifert matrix from a Seifert surface

I read multiple example about how to find the Seifert matrix from a Seifert surface but I just don't see it. I have trouble with the "under" and "over" crossings when we find the different loops. Here is an example from the book "Knot Theory and its…
user2795243
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Is a knot in $\mathbb{R}^3$ that can be untied necessarily trivial?

Is a knot in $\mathbb{R}^3$ that can be untied necessarily trivial? Trivial means it is equivalent(of the same knot type) to a circle in a plane. A knot K is the image of a homeomorphism of the unit circle into $\mathbb{R}^3$. K is said to be…
陈通昱
  • 130
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Perko pair - What's the handedness of these pictures?

In 1974, a paper titled On the Classification of Knots by Ken Perko appeared showing that the knots $10_{161}$ and $10_{162}$ in Dale Rolfsen's knot table were actually the same knot. He included this picture, showing how to deform one into the…
Akiva Weinberger
  • 23,062
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Are the borromean rings a torus link?

The $(3,3)$ torus link produces three unknots that are pairwise linked, which makes me suspect that the borromean rings ─ which are otherwise rather similar ─ cannot be produced through any $(p,q)$ torus link combination, and therefore that there is…
E.P.
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School project in knot theory

Can someone suggest an idea for a school project in knot-theory for a 13 year old? Thanks
Prateek
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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 conventional knot diagrams are consistent with the…
Joseph O'Rourke
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Knot theory: showing that the ambient isotopy relation is symmetric

My apologies for this rather elementary question, but here goes: I didn't have any trouble figuring out how to prove that the 'standard' homotopy/isotopy definitions give rise to an equivalence relation, but I've been struggling trying doing the…
mval
  • 353
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Knots with Fox tricoloring number $\mathrm{tri}(K)=27$

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 succeed.
Amira_Shikhi
  • 101
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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 positive integers? I have found the general form of…
wilsonw
  • 1,016
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What's the best material for making knots?

Just a week ago I started to study knot theory. I have been using copper wire to make knots, but it is too hard and sharp. Strings are so soft that they cannot maintain their shapes. It's difficult to put string knots up. What else can I use to make…
jotae
  • 61
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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) $H_2=0$, and I saw somewhere 4) the knot group…
yoyo
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Can the HOMFLY polynomial be obtained from the Kauffman Polynomial for torus knots?

This is essentially a yes/no/reference request question. Let me first just ask my question: Is there a known relationship between the HOMFLY and Kauffman polynomials of torus knots? In particular, is there a published result which says that given…
John L
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