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Questions tagged [knot-invariants]

For properties of knots that remain unaffected by Reidemeister moves

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Understanding Milnor $\bar{\mu}$-Invariants

I find Milnor's $\bar{\mu}$-invariants a bit confusing. My understanding of their calculation is as follows: draw the link diagram, label the arcs write down Wirtinger presentation of the link group $...
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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?
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Where can I find a proof that the crossing number of a knot/link is a knot invariant?

Where can I find a proof that the crossing number of a knot/link is a knot invariant? I know that this is in fact a true statement when you consider that the presentation of the knot is the one with ...
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Computing the Cohomology of Knot Space vs Cohomology of Non-compact Knots

In Vassiliev's paper of the cohomology of knot spaces, he considers the long-knots or non-compact knots i.e. embedding of $S^1$ into $S^3$ that go through a fixed point. He claims that this is meant ...
amd1234's user avatar
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Power series expansions and limits of knot invariants

I move the question here Background(ignore this paragraph if you know finite type invariants well): Recall that a finite type invariant of degree $n$ is an invariant $V$ such that $V^{(n+1)}=0$ where $...
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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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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. ...
Alex's user avatar
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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 ...
guest's user avatar
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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 $v$ ...
Johanna Hirvonen's user avatar
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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, ...
dfg dfg's user avatar
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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 $V_L(t) \to V_L(t^{-1})$. ...
hopftype's user avatar
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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 ...
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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(...
ParabolicX's user avatar
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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 ...
Philippe Knecht's user avatar
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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 ...
3299792458777's user avatar
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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 ...
Philippe Knecht's user avatar
3 votes
1 answer
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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 ...
Eric Ley's user avatar
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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 ...
horned-sphere's user avatar
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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 ...
Juan Felipe Salamanca Lozano's user avatar
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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 ...
Philippe Knecht's user avatar
6 votes
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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 ...
Sidharth Ghoshal's user avatar
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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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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? (...
Esther Jacob's user avatar
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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 ...
Teddy Astor's user avatar
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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 ...
Juan Felipe Salamanca Lozano's user avatar
1 vote
2 answers
176 views

Determinant of Alexander Matrix for Torus Links

The core of the problem Let $q,r\in\mathbb N$ be natural numbers with greatest common divisor $d$. Consider the $(q-1)\times(q-1)$-matrix $$ B:=\begin{pmatrix} -1\\ 1&-1\\ &1&-1\\ &&...
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Acyclic complex in Khovanov homology

I am reading Dror Bar-Natan's paper Categorification. In section 3.5.1 (page 9), "Invariance under R1", it is claimed "It is easy to check that $\mathcal{C}'$ is subcomplex of $\...
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Struggling to Show Alexander Polynomial is a Knot Invariant Using Skein Relations

For (one of) the books I am using to learn knot theory, the Alexander polynomial is defined by the skein relation, or the unknot has polynomial 1 and the relation $\Delta(L_+)-\Delta(L_-)+(t^{1/2}-t^{-...
junglekarmapizza's user avatar
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Wirtinger's presentation gives that the link group of two disjoint circles is $\mathbb{Z}^2$?

Given two disjoint circles $a$ and $b$ that are projected in a way so that there's a positive and negative $a$-over-$b$ crossing, Wirtinger's presentation gives that the generators $a$ and $b$ commute ...
Victor's user avatar
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Jones polynomials of alternating knots

Fox's Trapezoidal Conjecture asserts that the coefficients of the Alexander polynomial of an alternating knot alternate and the sequence of their absolute values forms a trapezoidal shape. The same is ...
Adam's user avatar
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How to tie a tight knot around my cable?

I would like to tie my new connector to my cable tightly using the provided string. This is what the untied knot and the the tied knot look like; unfortunately, the adaptor can still slip and slide ...
Mo Pro's user avatar
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4 votes
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Kaufman Bracket on Links vs Framed Links

The book "Quantum Invariants: A Study of Knots, 3-Manifolds and Their Sets" by T. Ohtsuki gives the following definitions: A framed link is the image of an embedding of a disjoint union of ...
user1104937's user avatar
2 votes
1 answer
103 views

How to recognize a knot, practically?

This question was inspired by the very first exercise in Thurston's Three Dimensional Geometry and Topology, where he gives a picture of a very tangled up loop and asks what manifold it depicts. I ...
KJL's user avatar
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How does KnotInfo classify knots based on DT Code?

The site KnotInfo can classify many small knots up to mirroring given a Dowker–Thistlethwaite Code of the knot, and can even identify whether a knot is a connected sum of other small knots. Being very ...
n3rl's user avatar
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Why should boundary maps be degree 0 in Khovanov Homology?

In Bar-Natan's On Khovanov's categorification of the Jones polynomial (https://arxiv.org/abs/math/0201043), the claim in section 3.2 when constructing the differential $d_\xi$ is that $d_\xi$ ought to ...
failedentertainment's user avatar
3 votes
1 answer
98 views

Jones polynomial of a knot in terms of its Seifert matrix

It is well known that the Alexander polynomial of a knot can be written in terms of the Seifert matrix of the knot by a simple relationship $$\Delta(t)=\det(V^T-tV),$$ where $t$ is a formal variable ...
Ramiro Hum-Sah's user avatar
2 votes
1 answer
100 views

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_+ ...
Nobilis's user avatar
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1 vote
1 answer
105 views

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 ...
Léo Mousseau's user avatar
1 vote
1 answer
80 views

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 ...
Alex's user avatar
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4 votes
1 answer
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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 ...
Haldot's user avatar
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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. ...
the's user avatar
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2 votes
1 answer
147 views

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 ...
Apoorv Potnis's user avatar
3 votes
0 answers
60 views

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 ...
Léo Mousseau's user avatar
2 votes
0 answers
78 views

If L=L1#L2 is a positive link, are L1 and L2 also positive links? What about adequateness?

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 ...
Léo Mousseau's user avatar
1 vote
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50 views

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 ...
berto's user avatar
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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 ...
HXR's user avatar
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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 $K$ ...
WhenYouHaveNoClue's user avatar
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32 views

Does the Gordian distance depend on the chosen link diagram?

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 $L$. ...
WhenYouHaveNoClue's user avatar
0 votes
1 answer
49 views

Non ambient isotopic knots with same Seifert matrix

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 ...
WhenYouHaveNoClue's user avatar

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