# Questions tagged [knot-theory]

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

970 questions
Filter by
Sorted by
Tagged with
13 views

### 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 ...
56 views

### 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 ...
37 views

### 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 ...
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 ...
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 ...
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 ...
29 views

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 ...
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....
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 ...
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!
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 ...
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 ...
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 ...
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 ...
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 ...
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?
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!
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 $h\colon X\to X$ ...
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 $p,q$ ...
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 ...
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 ...
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 ...
93 views

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 ...
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 ...