Questions tagged [knot-theory]

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

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A Certain Degree 12 Covering of the Figure-8 Complement

Let $M=S^3\setminus K$ be the figure-8 knot complement. The fundamental group of $M$ has a presentation $\pi_1 M=\langle x,y\mid x^{-1}yxy^{-1}x^{-1}y^{-1}xyx^{-1}y^{-1}=1\rangle$. There is a ...
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Notes on Low-Dimensional Topology

I am studying algebraic topology at the moment and I'm halfway done with Hatcher's book. I am extremely interested in low-dimensional topology, so I was wondering if anybody knows a good set of notes ...
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cusp shapes from vertex invariants

I am looking for an algorithm for computing the cusp shape from the vertex invariants of a complete triangulation of the toroidal cusp neighborhood of a knot complement. I have spent a fair amount of ...
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1answer
60 views

Null-spaces modulo n

Let A and B be square integer-valued matrices (possibly of different size) such that their null-spaces modulo n (i.e. the set of vectors v that satisfy Av = 0 mod n) are isomorphic (i.e. there exists ...
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32 views

Computation of Colored HOMFLY Polynomials

I am trying to understand the colored HOMFLY polynomials. The theoretic description Anna Aiston gave in her PhD thesis is really nice, but what about the computation? I would like to understand the ...
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36 views

Introductory book on knot theory in smooth category

I know several introductory books on knot theory in PL category, but I'm not familiar with PL topology. So I'm looking for introductory books on knot theory in smooth category. Could you recommend ...
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29 views

Why is ambient isotopy the preferred invariant of knots as opposed to isotopy?

Why is ambient isotopy the preferred invariant of knots as opposed to isotopy? From wikipedia: Two mathematical knots are equivalent if one can be transformed into the other via a deformation of $\...
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39 views

Application of higher-dimensional knot theory

There are quite a few applications to classical (1-dimensional) knot theory, such as knotted proteins, DNA,... Are there any potential applications of 2-dimensional knot theory? E.g. knotted surfaces ...
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2answers
60 views

Calculating homology groups of a generic knot

I am trying to calculate the homology groups of a knot embedded in $S^3$. This is what I have so far: $$H_0(S^3-K)=\mathbb{Z}$$ since $S^3-K$ is path connected; from Alexander duality I have that: $$...
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1answer
24 views

Calculating higher homotopy groups of (complements of) knots

There are techniques to calculate the group of a knot, i.e. the fundamental group of its complement in a manifold, but are there techniques to calculate its higher homotopy groups? Can anyone suggest ...
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1answer
44 views

Linking number and Homotopy

This statement has been taken from Wikipedia page. " homotopy classes of a curve in 3-space minus a circle are determined by linking number. It is also true that regular homotopy classes are ...
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1answer
50 views

Characterisation of longitude and meridian of Knot groups

$V = V(k)$ denotes a tubular neighbourhood of the knot $k$ and $C = S^{3}− V$ is called the complement of the knot. $H_{j}$ will denote the (singular) homology with coefficients in $\mathbb{Z}$ ...
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Recommendation for Low-dimensional topology textbook [closed]

Can anyone here recommend a low-dimensional topology textbook that contains knot theory and 3,4-manifolds?Or should I look for these subjects in separate textbooks?
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The Alexander polynomial of the theta graph

I am trying to figure out Alexander polynomials, so I am trying to calculate some that look simple. I started with the theta graph which has presentation $$\langle a,b,c|c=ab\rangle$$ I have ...
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1answer
45 views

The $p$-completion of a knot group

In Serre's LNM 5, in the section on profinite groups, he give the exercise: Let $k$ be a knot in $ \mathbb{R}^3$, and let $G= \pi _1( \mathbb{R}^3-k)$ be the knot group of $k$. Show that the $p$-...
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Analogy to Van kampen theorem for Quandles

This is from thesis of David joyce, An Algebraic Approach to Symmetry with Applications to Knot Theory Please help me to understand the $\lim$ $AQ(U_{i}$,$U_{i} \cap K) $as given on page 42 of the ...
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2answers
442 views

Trefoil knot cannot be injectively projected to a plane?

I'm trying to do the following geometry qual problem from my university: Give an example of a smooth embedding $f:S^1\rightarrow\mathbb{R}^3$, such that for each plane $P$ in $\mathbb{R}^3$, $\pi_P\...
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45 views

linking form of a knot in terms of Seifert matrices

I have troubles understanding the linking form and need some help with it. What I understand so far is the following: For some $2n+1$ dimensional rational homology sphere $M$ we can consider the ...
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35 views

When is a clasper a tame clasper

My reference for this is Habiro, Claspers and finite type invariants (2000), https://projecteuclid.org/journals/geometry-and-topology/volume-4/issue-1/Claspers-and-finite-type-invariants-of-links/10....
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1answer
109 views

Wrapping a Rubber Band around a Cube

Imagine a cube wrapped with string so that each of the six sides has two linked "L's" on it, like this image (or some reflection or rotation thereof). You can get a lot of different knots ...
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32 views

Braid words for 2-bridge knots? (reference request)

I was hoping that somebody could point me towards a reference where I could learn about braid word representations of 2-bridge knots. Thank you!
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1answer
31 views

Knots with infinite crossing

For me knots are embedding of $S^1$ in $\mathbb{R}^3$. I have following questions: Will knots have infinite crossing? If so, Why are we considering only knots with finite crossing ? Can someone ...
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1answer
150 views

What is the degree of an n-fold branched cover over a trefoil?

The order-2 cyclic branched cover over a trefoil has degree 6, meaning the preimage of any point off the trefoil has cardinality six. (You can find a wonderful video of this here, made by Moritz ...
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1answer
37 views

Connected sum of a link and a knot

Let $L=K_1 \cup K_2$ be a two-components link in a copy of $S^3$ and let $K$ be a knot, thought in a different copy of $S^3$. In other words, we have two couples $(S^3, L)$ and $(S^3, K)$. Let us ...
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51 views

Does snappy undertstand Gauss codes?

The documentation says it does, but experiment says it does not. For example, this: N=Manifold('Gauss: [(1,-2,3,-4),(5,-1,7,-3),(4,-5,2,-7)]') Errors out. Am I ...
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1answer
1k views

Topologically, what is a 'string' from string theory?

To begin: I am not a crank. I am not sure how well-founded my titular question is, but it was interesting enough that I decided to bring it to MSE. For context: I am an undergraduate mathematics ...
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1answer
42 views

Parametrization of a knot isotopy

I am working on a computer visualization of a knot isotopy of the standard unknot embedding to an unknot with a Reidemeister I move. Does anyone have a formula?
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38 views

Easy to use program to calculate the HOMFLY polynomial of a braid word?

Easy to use program to calculate the HOMFLY polynomial of a braid word? Looking for a reference. Thanks!
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1answer
43 views

Lifting a map to a homeomorphism of coverings

This is a part of the proof of Lemma 4 of "Cobordism of classical knots" by Casson and Gordon. Here, $\widetilde{X}$ denotes a prime-fold cyclic covering of $X$. Let $h\colon X\to X$ ...
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38 views

Are solvable links allowed to be split?

We define a solvable link as a link that can be built up iterating cabling and connected sums from the unknot in $S^3$. I recall the definition of cabling: a $(p,q)$-cabling of a knot $K$, where $p,q$ ...
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43 views

Knots inside loops

Reading about practical camping, sailing... knots, you find some of them can be done without having access to the rope's ends (if you don't require it to be attached to another loop, ring, etc.). That ...
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1answer
47 views

Questions on the symmetry of of the alexander polynomial and the rank of the Seifert matrix

For my bachelor thesis I am using the book "Lectures on the topology of 3-manifolds. an introduction to the Casson invariant"(1999) by Nikolai Saveliev. Regarding the Alexander Polynomial as ...
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1answer
48 views

Definitions of Knot Theory

I am currently doing a course in Knot Theory and after looking at different texts I have found many ways to define knots and knot equivalence. In our course we are given the following definitions: A ...
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1answer
93 views

Linking number of torus links

Let $(p,q)$ be a pair of coprime integers; the torus knot $K(p,q)$ is the unique (up to isotopy) curve on the boundary of a solid torus which is homologous to $p\cdot \lambda + q \cdot \mu$, where $\...
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1answer
44 views

Making precise Dehn filling

Dehn surgery along a knot is a well-known construction: choose a regular neighbourhood $N(K)$ of a knot $K \subset S^3$, let $X_K := S^3 - N(K)$ and choose an essential simple closed curve $\alpha$ on ...
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108 views

Knot theory and creative writing?

I am a Ph.D. Candidate in Creative Writing and an M.S. Student in Mathematics and I'm writing my master's thesis on knot theory and trying to tie in applications to creative writing. Has anyone come ...
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1answer
83 views

Finding my mistake in a “proof” that there is no non-trivial knot

I am working on a presentation about the complexity of determining whether a knot is trivial, and while reading about some topological stuff I somehow managed to "show" that there are no non-...
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1answer
56 views

Alexander polynomial of some 2-component links

I would like to understand whether the multivariate Alexander polynomial of a link $L$ does not vanish everywhere for a certain class of links; I don't know the links' diagrams in general but I have ...
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1answer
58 views

Intuitive Question about Klein Bottles and Torsion in Surface Groups

This is just a geometric sort of question about how to picture/intuit on homology, hope it's considered attractive and not just "vague" haha So if you take a typical generator for the ...
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24 views

Can a diagram of knot with minimal number of crossings contains a degenerate crossing or cancelling pair of crossings?

Given a knot diagram $D$ and a quandle $Q$. One can associate an element of $Q$ to an arc $\alpha$ of $D$, which called the color of $\alpha$ and denoted by $c(\alpha)$. By giving a color to every arc ...
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1answer
48 views

Knot being a subset of another knot.

Can you show an example of a nontrivial knot $ K $ in $ \mathbb{R}^{4} $ such that $ K \cap \mathbb{R}^{3} $ is a nontrivial knot in $ \mathbb{R}^{3} $ ? It would be nice if you used parametric ...
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1answer
80 views

Can Isotopic Arcs be Isotoped So That They Don't Intersect Until $t = 1$?

Suppose $A, B$ are two disjoint arcs in $X = \mathbb{R}^3$. An 'arc' is a homeomorphic image of $[0,1]$. Let $F$ be an ambient isotopy on $X$ carrying $A$ to $B$ - that means there is a ...
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1answer
29 views

Does the factorization operation for composite knots applicable for any projection of the knot?

Let $K$ be a knot in 3-space. We know that if $K$ is not prime (composite) then $K$ can be represented as a connected sum of two knots, both are non-trivial. My question is : does this apply to any ...
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1answer
50 views

Prerequisites for Neuwirth's 'Knot Groups'

I have never studied knot theory before. I would like to get into the subject. I am interested in studying knots from a topological perspective (as opposed to a combinatorial one.) I am studying knot ...
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48 views

Is this knot-forming process a categorical construction?

Every embedded planar graph gives rise to a link (collection of knots) according to a particular procedure (I've heard it called the Mercat procedure). I am trying to determine whether this ...
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1answer
58 views

Inverse of Trefoil Knot Paramaterization

I'm working on a coding project at the moment where I have to integrate a function around a trefoil knot. I'm using the following parametrization to do so. $x = \sin(t) + 2\sin(2t)$ $y = \cos(t) - 2 \...
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22 views

Formal definition of a knot diagram, including data for the crossings

The definition of a knot diagram has, in some sense, two parts: the 2-dimensional projection itself, packaged with the corresponding data about where the over- and under-crossings are. It's easy ...
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82 views

Suppose a theta graph's three cycles are all unknots. Must it be unknotted?

Here's small knot theory question involving a knotted graph. A theta graph is a $\theta$ shape: two vertices with three parallel edges between them. It has three cycles, each obtained by deleting an ...
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1answer
50 views

Number of prime knots for crossing number greater than 16

How many prime knots are there for crossing numbers greater than or equal to 17? I can't seem to find any useful resources indicating that they have been computed or that algorithms exists to compute ...
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1answer
86 views

Is number of knot strands an invariant?

Question: Does the number of components in a knotwork depend on the particular planar embedding? I've been investigating how to compute the number of components ("separate strands") in a ...

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