# Questions tagged [knot-theory]

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

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### How to count the number of line segments in a knotwork?

Question: Given an embedded planar graph $G$, how can you compute the number of segments in the corresponding knotwork? Does the number of segments depend on choice of embedding? Background Any ...
44 views

### Alexander Polynomial

Recently I learned the Alexander Polynomial of a knot and how to find the polynomial for a given knot. Now there are some questions arise, I am trying to give some classification of hyperbolic knots, ...
1 vote
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### How to understand the framing of a knot?

I was told the framing of a knot is the linking number of the push-off. But I don't understand why the framing does not depend on the knot but only on the parallel copy. How about a Legendrian knot? (...
1 vote
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### Tied down chess pieces forming knots

I want to make a public chess table, with every piece tied down securely so it cannot be stolen. Each of the pieces have attached one arbitrarily long thin steel cable, with one end in the side of the ...
1 vote
24 views

### Does every knot have a diagram such that every arc is a bridge?

Since the number of crossings in a knot diagram is the same as the number of arcs (or edges), can you construct a diagram of a knot where each arc crosses over exactly one other arc? My first thought ...
1 vote
37 views

### Jones polynomial in Bar-Natan’s paper “On Khovanov’s Categorification of the Jones polynomial”

I’m reading Bar-Natan’s paper “on Khovanov’s categorification of the Jones polynomial”, I had previously been reading Lickorish’ book to have a good understanding on the Jones polynomial before diving ...
65 views

### Computing Floer homology of knots, going from graph to homology (following Manolescu's high-school level slides)

Recently I came across the following expository slides of Ciprian Manolescu: The unknotting problem: a journey from elementary to advanced mathematics, talk for the high school students at the ...
1 vote
45 views

### Do knots behave the same way if they are defined to have "endpoints at infinity"?

Typically, knots are defined as embeddings of $S^1$ into $\mathbb{R}^3$ or $S^3$. However, in real life, knots usually sit in the middle of a long rope, whose endpoints you might as well model as ...
1 vote
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### Jones polynomial of the left-handed Trefoil knot - which crossing for skein relation L_0?

I tried computing the Jones polynomial for the left-handed trefoil knot, but ran into a bit of an issue with how I pick my crossings for the L_0 skein relation. I decided to work with the lower L_+ ...
59 views

### Standard fact in knot theory

In "Knots and Links in Spatial Graphs" (Journal of Graph Theory, vol.7 1983, 445-453), Conway and Gordon write : "Now it is a standard fact in knot theory, not hard to prove, that any ...
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### Knots and links of parallel arcs

Starting for this diagram, we may say the following: $a)$ It can be coloured $\mod3$ since: @blue: $0+1 \equiv (2*2) \mod3$ @red: $2+0 \equiv (2*1) \mod3$ @green: $1+2 \equiv (2*0)\mod3$ $b)$ Here we ...
46 views

### Genus 1 fibered knots in an integer homology sphere.

It is well known that genus 1 fibered knot in $S^3$ consist of trefoil and figure-eight knot. My classmate tells me that genus 1 fibered knots in an integer homology sphere must be trefoil or figure-...
1 vote
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### Whitehead double of a non-trivial knot is non-trivial

How can one show that if a Whitehead double of a knot is trivial, then the original knot must have been trivial? Since the Alexander polynomial and the signature vanishes, and since it is not clear ...
32 views

### Is it possible to model a rope that splits and then joins again as a knot or link?

Context I'm trying to learn about knot theory so that I can use it to systematically solve a disentanglement puzzle I own. I have learned about how to model knots in a braid diagram, how to write that ...
Here is the question I am trying to solve: Let $X$ be a based space, and let $PX = \{ \beta: I \to X | \beta(0) = *\}.$ Show that $p_1: PX \to X$ by $p_1(\beta) = \beta(1)$ is a based fibration. I am ...
When introducing topologically slice knots (i.e. knots $K\subset S^3=\partial D^4$ which bound a locally flat disc in $D^4$) one explains the local flatness condition by noticing that without local ...