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Questions tagged [knot-theory]

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

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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 $...
Mathemann's user avatar
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1 answer
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Proving the Uniqueness of the Method for Reducing Rational Tangle-Numbers to Zero from Conway's Rational Tangle

I am a high school student working with Conway's rational tangles and specifically with reducing tangle-numbers to zero using T and R operations. (I decided to not use the continued fraction form in ...
Saeol Ahn's user avatar
2 votes
1 answer
42 views

Example where rotation number of Legendrian knot depends on choice of Seifert surface

In Surgery on Contact Manifolds and Stein Surfaces there is the following exercise [below $Y$ is a 3-manifold, and $e(\xi)$ is the Euler class of $\xi$]: Exercise 4.2.6. Find a contact structure $(Y,\...
Hrhm's user avatar
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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 ...
Zuriel's user avatar
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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 ...
fxxxxx's user avatar
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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 ...
H-a-y-K's user avatar
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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 ...
ripan sharma's user avatar
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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?
Tutusaus's user avatar
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Linking number of irregular curves

Let $f,g \colon S^1 \to \Bbb R^3$ be two continuous functions (not necessarily embeddings) whose images are disjoint. We define the linking number of these closed curves to be $$ L(f,g) = \text{deg}(...
BigbearZzz's user avatar
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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 ...
Tutusaus's user avatar
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Requirements for an Alexander matrix to represent a knot diagram

Could there exist a finite set of requirements for an Alexander matrix such that if and only if these requirements are met, that the Alexander matrix represents a knot diagram. For example, i am quite ...
dfg dfg's user avatar
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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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Why $H_3(S^3)\to H_2(T)$ is injective ? (Homology of knot complement )

I have a question about the homology of the complementary space of knots. I was able to find $H_2(X)$ and $H_1(X)$ ( $X\colon $ knot complement ) by using MV sequences, but I was not able to derive $...
fxxxxx'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 $...
Eric Ley's user avatar
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Compute the homology of this configuration space.

Let $U\subset V$ be finite labeling sets, and $K:\mathbb S^1\to\mathbb R^3$ be a knot. Consider the configuration space with points labeled $U$ lying on the knot, to make this space connected we fix ...
Eric Ley's user avatar
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an example of a knot diagram K such that K ~ O(unknot) and K requires a type 3 move to get equivalence started.

I was reading a book about knots: the book "on knots" by Louis Kaufman In one of the exercises he wrote: Give an example of a knot diagram K such that K ~ O(unknot) and K requires a type 3 ...
Hesam's user avatar
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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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1 answer
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Understanding the proof of Fenn-Rourke Theorem

Fenn-Rourke Theorem states that Framed links can be transformed into each other by Kirby moves if and only if they can be done by Fenn- Rourke moves. I'm trying to understand the proof of it in V.V....
user540663's user avatar
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Show that a given procedure generates all rationals greater than 1

Question without Context Let be given a sequence of natural numbers $(A_1,A_2,...)$, such that $A_i\ge A_{i-1}+1$, and consider the set generated by dividing $(1,...,A_1)$ by $1$, $(A_1,...,A_2)$ by $...
Stamatis'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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2 votes
1 answer
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Different statements about the peripheral system as a complete knot invariant

I am somewhat confused about the different flavours in which the statement "the peripheral (group) system is a complete knot invariant" usually comes, and I believe not all of them have ...
Minkowski's user avatar
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Peripheral subgroup determined up to conjugation

Let $K$ be an oriented knot in $S^3$, let $X_K := S^3 - \mathrm{int} \ N(K)$ be its knot exterior and let $i: \partial X_K \hookrightarrow X_K$ be the subspace inclusion. The peripheral subgroup is ...
Minkowski's user avatar
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2 votes
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Does a torus knot give a Seifert fibering of the 3-sphere?

Let $K$ be a $(p,q)$ torus knot on the torus $T_1$. Via the map \begin{equation*} H=\begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix} \end{equation*} $K$ becomes a $(q,p)$ torus knot on $...
Hempelicious's user avatar
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Automatic knot/link identification

My program spits out links as a crossing diagram. In fact, it does have some topological data (not only graphic) since the links are constructed as sums of tangles, and instead of drawing line A to ...
Hauke Reddmann's user avatar
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Show that the two-fold cyclic branched cover of a knot $K $ with trivial Alexander polynomial is a homology sphere.

I meet this when reading a paper about cyclic branched covering spaces of knots. There is no explanation nearby, can anyone tell me how to understand this?
Tsoshamry'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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0 answers
47 views

Why $\pm 1$ surgery on homology sphere again yeilds homology sphere?

Let $L$ be a framed link in an integral homology 3-sphere $M$. I read in this paper that if $L$ is algebraically split (pairwise linking number is zero) and unit-framed (framing $\pm 1$), then by ...
Eric Ley'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 ...
NadavS's user avatar
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1 answer
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Knot Theory: First Homology group of cyclic double cover of $S^3$ branched over a link L and nullity

According to Lickorishs book (Introduction to Knot Theory) I know that $|H_1(X;Z)| =$ order of the group $ H_1(X;Z) = |det(A+A^t)|$, where X is the double cover of $S^3$ branched over a link L and A ...
Koffie tante's user avatar
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0 answers
20 views

Minimal Morse link-derived presentation of a rational knot

Copied from the Knot Atlas and annotated. As you see, $7_2$ needs $4$ braid strands (any rational knot needs at most 4), but if we cheat and define "closure" differently, we could say "...
Hauke Reddmann's user avatar
2 votes
0 answers
175 views

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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1 vote
1 answer
74 views

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
0 votes
1 answer
50 views

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 ...
Eric Ley's user avatar
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2 answers
123 views

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
2 votes
1 answer
61 views

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 ...
horned-sphere's user avatar
1 vote
1 answer
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 ...
Robert Abramovic's user avatar
2 votes
1 answer
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 ...
user540663's user avatar
0 votes
1 answer
132 views

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 ...
user540663's user avatar
2 votes
1 answer
56 views

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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0 answers
37 views

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
1 vote
0 answers
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 ...
elvexo's user avatar
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1 vote
1 answer
89 views

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....
Eric Ley's user avatar
  • 738
1 vote
2 answers
77 views

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
90 views

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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0 answers
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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
1 vote
0 answers
76 views

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
0 votes
0 answers
45 views

(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.
Alexandra Fikiori's user avatar
0 votes
1 answer
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 ...
Philippe Knecht's user avatar
1 vote
1 answer
241 views

Prove that every knot diagram with two crossings is equivalent to the unknot.

I am a beginner studying knot theory, and we covered the Reidemeister moves on link diagrams in class today. The question in the title is the one I am struggling with now. I attached an image of the ...
cheddarscooper's user avatar
3 votes
1 answer
189 views

What is the algorithmic complexity of knot equivalence?

Question. Given two (tame) knots by their link diagrams, what is the algorithmic complexity (e.g. time in the size needed to store the link diagrams) to decide if the represented links are isotopic? ...
Thomas Preu's user avatar
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