# Questions tagged [knot-invariants]

For properties of knots that remain unaffected by Reidemeister moves

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### Characteristic class detecting "upward-facing" surfaces

Let $\Sigma \subseteq \mathbb{R}^3$ be a smoothly embedded compact oriented surface with boundary. Let $\vec{n}: \Sigma \rightarrow \mathbb{R}^3$ be the field of unit normal vectors associated to the ...
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1 vote
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### Proving that an algebraic (aborescent) link that has exactly one negative sign in its Conway notation has an almost alternating projection

I'm working on Exercise 5.32 in The Knot Book by Colin Adams, which asks to prove that an algebraic link that has exactly one negative sign in its Conway notation has an almost alternating projection. ...
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### Software packages to compute finite type invariants of Polygonal Knots

Assume I have a polygonal knot, $K$, represented as its set of vertices $\{\mathbb{v}_i| \mathbb{v}_i\in\mathbb{R}^3\}_{i=1,...,n+1}$, where $n$ is significant, let's say $100<n<500$. Which ...
• 51
1 vote
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### When is Morton-Franks-Williams inequality for knots strict?

I am reading Kawamuro's paper on Morton-Franks-Williams inequality (https://arxiv.org/abs/math/0509169). It says that a knot $K$ with braid index $b$ and maximal/minimal degrees of the variable $v$ ...
1 vote
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### Omitting the last relation in the Wirtinger presentation of a link group

In my knot theory class homework I encountered the following question: Prove that for every link, when calculating the Wirtinger presentation of the fundamental group of its complement, you can ...
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### Connection between different ways to calculate the knot determinant

I understand that there are multiple ways to calculate the knot determinant, one is through the Alexander polynomial, the other is by creating another matrix which uses the linesections and crossings, ...
• 21
1 vote
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### Chirality and Colored Jones Polynomial

It is well understood that the usual Jones polynomial of a knot or link can be related to the Jones polynomial of the mirror image of the knot/link by changing the variable $V_L(t) \to V_L(t^{-1})$. ...
• 11
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### Confusion on the Definition of Isolated Chord

An isolated chord diagram is usually defined to be a chord diagram with a chord that doesn't intersect any other chord. But in this notes, it is defined to be a diagram with a chord that relates two ...
• 738
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### What is the Jones Polynomial for the Borromean Link?

I was looking up the Jones Polynomial for a project I’m working on and came up with this equation from the knot atlas: $$-q^3-q^{-3}+3q^2+3q^{-2}-2q-2q^{-1}+4$$ However, I know that when entering VL(...
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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
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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 ...
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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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### 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 ...
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### Are there examples of different knots with identical Jones polynomials and different Seifert Genus?

I'm wondering if its ever possible to find two non-isotopic knots which have identical jones polynomials but different seifert genus? Attempting to google for this I found this example of non-isotopic ...
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### 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, ...
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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? (...
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### 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 ...
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### 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 ...
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_+ ...
• 127
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 ...
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### What kind of mathematical knot is the square knot?

What kind of knot is the square knot? I made an attempt to calculate the Connway notation of the square knot and it did not match with any of the prime knots. Either I did the calculation wrong or the ...
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### Proving that $n$-component Brunnian link is nontrivial

I stumbled upon the attached image. It shows a way to construct an $n$-component Brunnian link for any $n\geq 3$. That is, this link is not trivial, but deleting any of its components makes the new ...
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### When a 3-strand pretzel link is an unknot?

I want to find any (nontrivial) necessary (and also sufficient, if possible) condition to determine whether $(p,q,r)$-pretzel link is equivalent to the trivial knot, where $p$, $q$, $r$ are integers. ...
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### Is a complete knot polynomial known?

Wikipedia states in its article on knot invariant that Other examples are knot polynomials, such as the Jones polynomial, which are currently among the most useful invariants for distinguishing knots ...
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### Smooth 4-genus not agreeing with topological 4-genus intuition

I am aware that there are many examples of knots which have different topological 4-genus than smooth 4-genus (11n_34 is the one with the least amount of crossings, it is topologically slice but has ...
• 366
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Lets say I have an oriented positive link L (i.e. an oriented link which admits a diagram with only positive crossings) such that L=L1#L2 for two links L1, L2. Now my question is wether those two ...
• 366
1 vote
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### Which hyperbolic fibered knots have monodromy with a single singularity?

The figure eight-knot has pseudo-Anosov monodromy with no singularity. I have read that the (-2,3,7)-pretzel knot has pseudo-Anosov monodromy with a single 18-prong singularity on the boundary of the ...
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### Could anyone introduce some books about Knot Theory and polynomial?

I've already learnt basic topology, and understood part of differental manifolds, the embedding, and part of algerbraic topology, the simplicit homology. My boss advised me to finish a subject review ...
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### Every matrix $A$ with $\text{det}(A-A^T)=1$ can be realized as Seifert matrix

Let $A\in\text{Mat}(2n\times2n;\mathbb{Z})$ be an integer matrix with $\text{det}(A-A^T)=1$. I got told that every such matrix can be realized as Seifert matrix of a Seifert surface $F$ of a knot $K$ ...
Let $K,L:S^1\rightarrow S^3$ be two knots. The Gordian distance between $K$ and $L$ is defined to be the minimum number of crossing changes to convert a knot diagram of $K$ to a knot diagram of $L$. ...
I am looking for an example of two knots $K,J$ and Seifert surfaces $F,G$ of $K$ respectively $J$, such that for an appropriate basis both surfaces admit the same Seifert matrix, but $K$ and $J$ are ...