For questions on knot theory, the study of mathematical knots

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Equivalence (or not) of two Artin/Fox wild arcs

The repeating patterns in the wikipedia articles on wild arcs and wild knots seem to me to be not continuously deformable to each other. Is this true? For clarity, here is my diagram of the repeating ...
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30 views

Solving integrals with delta function constraints

What is the best way to solve integrals which use delta function constraints/restrictions? For example if I have the integral $\int_V ...
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1answer
60 views

Higher homology groups of infinite cyclic cover

Prove that all homology groups of the infinite cyclic cover of a knot complement are trivial except $H_1$. I've posted an answer below using Mayer-Vietoris. If you know of other arguments, please ...
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1answer
39 views

Showing that gluing two knot exteriors together contains subgroups isomorphic with the knot groups.

I'm working through Rolfsen's "Knots and Links" and section 9D exercise 10 has me stumped: Let $K_1$ and $K_2$ be knots in two separate copies of $S^3$ with respective meridians $m_1$ and $m_2$ and ...
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1answer
54 views

Definitions from topology

I'm reading some papers on the unknotting problem in Knot theory and am running into some notation I don't know (my exposure to topology is minimal, but I have seen it in Analysis courses, Algebra, ...
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2answers
70 views

Books about braid theory

I'm looking for books that talk about braid theory, in the sense of braid groups mostly, and not too advanced, if possible. With material understandable for an undergraduate. Thanks for any ...
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1answer
54 views

Background required to understand the mathematical definition of knots and their transformations

What are the concepts of math required as a prerequisite to understand Knot Theory? I'd like to be able to make a humble beginning by being able to mathematically define knots and the non-rigid ...
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60 views

What is a 3-cell? What is a 3-disk?

Checking the usual places on the Web doesn’t (right now) yield a short answer to this simple question. I’m worried that I will spend just as long trying not to confuse $n$- with $(n+1)$-cells as I did ...
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1answer
74 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 ...
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1answer
47 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 ...
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83 views

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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Linking of $S^p$ and $S^q$ in the $\mathbb{R}^d$ space

Can we have a nontrivial linking of a $S^p$ sphere and a $S^q$ sphere in the $\mathbb{R}^d$ space (or in the ${S}^d$ space)? I suppose that it can happen only if $p+q<d$. For example, we can have: ...
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17 views

colouring knot diagram and its mirror by the same colouring

Let $K$ be a knot diagram coloured by any quandle $X$. Let the colouring used be $C$. Reverse the orientation of $K$ to obtain the reverse of $K$, denoted by $-K$. Then is it possible to colour $-K$ ...
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1answer
33 views

1-surgery on the figure-eight knot: reference request

As far as I know, 1-surgery on the figure-eight knot gives ($\pm$) the Brieskorn sphere $\Sigma(2,3,7)$. However, is there a citeable source for this? Sometimes Thurston's notes are mentioned, but I ...
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Recovering knot crossing orientations from a Gauss code or Dowker notation

Some common representations of knots do not directly give the sign/orientation of each crossing. For instance, the trefoil knot has Gauss code -1, 3, -2, 1, -3, 2 and Dowker-Thistlethwaite code 4 ...
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1answer
22 views

property of quandle

Let $ Q $ be any quandle. Let $ y,k, w$ be elements of $ Q$. Is it true that if $ y*k=y*w $ then $ k=w $? I don't think so since the second axiom of the quandles states that for any two distinct ...
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64 views

Are there enough knots to cover $\mathbb{R}^3$? [closed]

Actually, several years ago I was in a short, introductory, course about knot theory, and my original question that I posed was: "can the knots be used to classify homeomorphims in $\mathbb{R}^3$?". ...
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1answer
40 views

Are knot complements prime 3-manifolds?

Well, that is basically the question. Is the complement of a knot, i.e. an smoothly embedded copy of $S^1$ in $S^3$ a prime $3$-manifold? Here I mean by prime: A connected $3$-manifold $M$ is prime ...
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15 views

unknotted $n$-dimensional knot

Let $n$ be any integer. An $n$-dimensional knot is an $n$-dimensional manifold embedded smoothly into $\mathbb{R}^{n+2}$. If it is homeomorphic to a disjoint union of $n$-spheres, then it is denoted ...
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35 views

Determining slice knots

Lately I have been thinking about slice knots. Is there any known effective procedure for determining whether a knot is a slice knot?
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38 views

Seeking guide for project.

I have to submit a project within 2 months for 4th semester(M.Sc). I wish to do it on knot theory, although I know little about it. My plan is to make it an introduction to the subject and to ...
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24 views

Skein relationship and Alexander polynomial

Given is the three link diagrams of Conway $L_{0},L_{+},L_{-}$ and the corresponding Seifert matrices $M_{0},M_{+},M_{-}$. Prove that ...
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1answer
24 views

Does the elementary knot move really preserve the orientation?

So in this picture, the first diagram changed to the third diagram by the elementary knot moves, but the orientations of the first and the third are different. I wonder if in $R^3$ the moves don't ...
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18 views

Is the Alexander Ideal of a Link always Principal?

It is known that the Alexander ideal of a knot (i.e., a link of one component) is always a principal ideal since any tame knot in $S^3$ has a square presentation (Rolfsen, D. Knots and Links, pp. ...
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27 views

Is the Ambient Isotopy relation (for embeddings) symmetric?

Definition: Let $X$ and $Y$ be topological spaces. Suppose that $Y$ is compact and Hausdorff. Let $f,g:X\to Y$ be embeddings. We say that $f$ is ambient isotopic to $g$ (denote $f\sim g$) if there is ...
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1answer
39 views

Is the closure of $[\sigma_1^2,\sigma_2^2]$ in $B_3$ equal to the Borromean rings?

Is the closure of $[\sigma_1^2,\sigma_2^2]\in B_3$ (the braid group with $3$ strings) equal to the Borromean rings? If yes, is there any simple proof?
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127 views

Generators of the braid group

Let $C$ be the plane curve in $\mathbb{C}^2$ defined as $\{x²-y^3=0 \}$. The fundamental group of $\mathbb{C}^2 \backslash C$ is the same of the trefoil knot : $\langle a_0,a_1 \; : \; a_0a_1a_0= ...
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22 views

Proof of the existence of knot which realizes given polynomial as Alexander poly.

The following theorem by Seifert is well known. A first half of the proof can be found in the book by Rolfsen (pp.171-172). I have questions that can be asked along the way, but the main question is ...
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7 views

t-minimal diagram of a surface in 4-space

Let D be a t-minimal surface diagram of a surface embedded in 4-space, i.e. D has minimal number of triple points over all possible surface diagrams. Is it true any double curve in D contains all ...
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31 views

Zeros of vector field extended from field lines

Suppose I have a finite number of 1-dimensional curves embedded in a 3-dimensional space. What can I say topologically about vector fields chosen so that these closed curves are field lines? For ...
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28 views

smoothing crossing point in knot diagram

For an oriented knot diagram, each crossing can be smoothed in two possible ways (we trace along one strand and before the crossing we move to the other strand) an oriented smoothing and an ...
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1answer
31 views

genus of knot and crossing number invariants

Genus of knot is defined to be the least genus among all Seifert surfaces of knot. Crossing number is the minimal number of crossings over all possible diagrams. Both genus of knot and crossing ...
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1answer
44 views

additivity of crossing number of composite knots

Let $K_1 \# K_2$ denotes the connected sum of tow knots $K_1$ and $K_2$. The crossing number is the minimal number of crossings among all knot diagrams, denoted by $c(K)$. It is conjectured that ...
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What is the relation between the construction of Reshetikhin-Turaev and Witten's paper?

I have read that the papers of Reshetikhin and Turaev provide a mathematically rigorous framework for the ideas contained in Witten's paper "Quantum field theory and the Jones polynomial". Can ...
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1answer
157 views

Software for drawing braid-related graphs

I am looking for a (preferably free) software that can draw braid-related graphs, such as and (Quoted from A Study of Braids By Kunio Murasugi, B. Kurpita) I have seen this question, but the ...
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1answer
25 views

How to construct a concrete isotopy from figure eight knot to its mirror image?

We know that figure eight knot is amphicheiral. But how to show that in terms of Reidemeister moves? Just several instructive hints will help a lot. I really have no idea..
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Is every $S^1$ knot orientable?

let $K$ be an $S^1$ knot, i.e., $K$ is a homeomorphic copy of the "standard" $S^1$ living in $\mathbb R^3$ ). I am trying to see if $K$ is orientable. This is what I have: 1) If the map sending $S^1$ ...
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47 views

if a circle is deformation retract of a manifold; then is there a knot deformation retract to an embedded M in 3-space

Let M be a metric space such that the circle is deformation retract of M. If a circle is embedded in 3-space, we obtain a knot. Suppose the space M is embedded in 3-space and its projection image in ...
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27 views

fundamental group of ribbon knot

The ribbon knot is a boundary of self intersecting disk such that the singularity is a self intersection of the disk along an arc. My question is that how to compute the fundamental group of ribbon ...
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22 views

Smoothing multiple crossings

Can you smooth multiple crossings at once in a knot when computing the bracket polynomial? If so is a there a rule for this?
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1answer
26 views

Bracket polynomial of a hopflink

When you are computing the bracket polynomial of the (for example) hopflink, why can you not smooth all the crossings in one go? Why do you have to only first start with one crossing? For example in ...
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14 views

Linking number and writhe of the hopflink

If you have two circles linked up to give the hopflink. Both these circles have clockwise orientation. Is the linking number and writhe going to be zero?
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1answer
38 views

Reidemeister infinity move

If you have a knot K. How do you apply a R infinity move to it? Is their an algorithm which tells you what to do at certain parts on a knot?
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5answers
374 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 ...
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1answer
32 views

embedding projective plane in 4-space? [closed]

Is it possible to embed projective plane in 4-space? If not what is the reason and what is the smallest singularity set?
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Compatible seifert circles

When are two seifert circles compatible? I originally thought that it was just a matter of looking at their orientations to check but it seems that is not enough.
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Is there a complete Link Invariant for links with N crossing.

Are there known examples of pairs $\left(f, N\right)$, where $f$ is a link invariant that is known to be complete when restricted to link diagrams that have at most $N$ crossings? (Ideally, f should ...
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1answer
77 views

math in horseshoe puzzle

We know that Rubik's Cube is a good demonstration of group theory. Correspondingly, for the horseshoe puzzle as in the picture below, is there a math language for it? Does it demonstrate any math ...
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43 views

Is it trivial that you cannot get apart 2 linked unknots?

I have read that: a trefoil is not an unknot is not trivial in knot theory. And it put the following question in me. Is it also not trivial that you cannot get apart 2 linked unknots? I mean, is it ...
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2answers
39 views

Knots isotopic to their mirror image

How do you prove a knot is achiral? Do you just show swap the under crossings and the over crossings?