# Questions tagged [knot-theory]

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

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• 3,405
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### Braid form of general link embedded in a genus $g$ surface?

It is known that a general braid word for a $T(m, n)$ torus link is $(\sigma_1\sigma_2\cdots\sigma_{n−1})^m$; for example, see here. Does anyone know if there is a braid form for a general link ...
• 5,451
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### A few questions on bracket polynomial

I have a few questions about bracket polynomials. We have the above equation, but when calculating the value of the figure D for a given link L, does the order in which this equation is applied ...
• 95
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### Achieving parallel wires with 180-degree rod rotations in an interconnected node system

We have 5 nodes on the left and 5 nodes on the right. You can imagine it this way: the nodes are on parallel rods and are connected with wires so that initially the wires are all parallel. We'll call ...
• 729
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### Markov Theorem for Braids

I was reading the Markov Theorem for Braids from the book "Braid Groups" by Kassel and Turaev. I am having a hard time reading the proof, given the length of the proof. Can anyone suggest ...
26 views

### Unknotting number

How is the unknotting number a knot invariant? I mean, if I have two links which are ambient isotopic do they have the same unknotting number?
• 657
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### Deducing inequality from exact triangle in Heegaard Floer homology

In Hom's lectures on Heegaard Floer homology, pages 8 and 9 contain a proof that $rk \widehat{HF}(Y) \geq |H_1(Y, \mathbb{Z})|$ for rational homology spheres. The proof involves using an exact ...
1 vote
101 views

### Is every non-intersecting closed loop about the origin with winding number 1 ambient $\mathbb{R}^{2}$-isotopic to the unknot?

Let $\gamma(t): [0, 1] \rightarrow \mathbb{R}^{2}$ be a counterclockwise-oriented closed loop that does not intersect itself and does not intersect the origin: $\gamma(0)=\gamma(1)$. Further, assume ...
103 views

### References needed for Dehn surgery and Kirby calculus

I learned from Colin Adams's book, $\textit{the knot book}$, that every compact connected three manifold comes from Dehn surgery on a link in $S^3$, and if two different Dehn surgery yield the same ...
• 327
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### a question in the proof of type $(p,q)$ torus knot and type $(q,p)$ torus knot are equivalent

a proof says that if we remove a disk which doesn't touch the knot from the torus, deform it to get a surface $S$ which consists of two bands attached to each other, then we can turn each band inside ...
• 327
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### 2-bridge knot with straightened strand

Apparently, every 2-bridge knot can be drawn such that of the four strands in the braid word, one strand remains straightened and is not crossing any of the other strands. Is there a general algorithm ...
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### Showing unknottedness with relative maximum and minimum

I want to prove (Rolfsen) If an embedding $e: S^1 \rightarrow \mathbb{R}^3$ has only one relative maximum and minimum in the $z-$direction, then $e(S^1)$ is unknotted. In particular, I do not have a ...
1 vote
118 views

### Triquetra proportions and Golden Mean?

I've been tinkering with drawings of Triquetras (triangular figures composed of three overlapping arcs), and wondering why this ancient symbol has appeared in so many cultures over so many ages. It ...
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1 vote
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### How to show this surface is compressible

This is exercise 4.12 in The Knot Book. Show that the surface in the figure is compressible by finding a disk that intersects with the surface in the boundary which bounds no other disk in the surface....
• 738
1 vote
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### Does the determinant of a Knot bound the number of primes for which the mod p rank is nonzero?

Definition. If $V$ is a Seifert matrix for a $\operatorname{knot} K$, then the determinant of $K$, denoted $\operatorname{det}(K)$, is the absolute value of the determinant of the symmetrization of ...
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### Why the bridge index of $8_{10}$ is 3?

This might be too elementary. I tried to deform the projection but couldn’t be able to find a projection of knot $8_{10}$ with 3 maximal overpasses. Is there any elementary reference on calculating ...
• 738
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### Is there an ill-embedded ball in the 4-sphere?

In https://arxiv.org/pdf/2102.04391.pdf, there is an explanation of how one could theoretically use a pair of knots $K$ and $K'$ (one slice and the other not) with the same 0-surgery to generate a ...
1 vote
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### Almost all knots are non invertible

K. Murasugi mentions in p.45 of his book "Knot Theory and Its Aplications" that almost all knots are non invertible, meaning that they are not equivallent to their reverses, where the ...
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### (p,q) and (q,p) torus knots are isomorphic

Can anyone help me understand why these 2 torus knots are isomorphic ? i tried to prove it using the parametric equations of a torus knot, but it didnt help.
67 views

### The degree of the Alexander polynomial is at most twice the genus.

The genus $g(K)$ of a knot $K$ is the minimum possible (topological) genus $g(S) = \frac{2-\chi(S)-B}{2}$ of a Seifert surface $S$ for the knot $K$ (where $\chi(S)$ denotes the Euler characteristic of ...
1 vote