Questions tagged [knot-theory]
For questions on knot theory, the study of mathematical knots and their properties.
1,190
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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 ...
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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? (...
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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 ...
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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 ...
2
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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 ...
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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 ...
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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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Algebraic sum of the transverse intersection points
On page 61 of An introduction to knot theory by Lickorish, second paragraph, the author used the term “algebraic sum of the transverse intersection points”.
I’ve never seen this when I read about knot ...
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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^{-...
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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 ...
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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 ...
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Natural candidates for energy function of knots in $S^3$?
Let $K\subseteq S^3$ be a knot.
In real-life, knots (like protein chains) $K$ moves around stochastically, and experimentally the lowest energy/highest entropy states are particularly simple from a ...
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Check if there is a knot between piecewise linear line connecting finite number of points
I have a finite number of ordered points connected by a piecewise linear curve, which connects all the points in order and then the last with the first one. Can I check if there is a “knot” in this ...
- 4,843
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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 ...
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How to generate tangles?
In my research I need to generate tangle diagrams for with small number of crossings ($\leqslant 10$).
By tangle I mean 2-tangle, that is an embedding of two arcs into a ball where endpoints of arcs ...
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Ray Tracing In Mathematical Spaces
I really enjoyed the Not Knot video, but I don't fully understand the mathematics which is going on there.
They are animating the space of the complement of the Borromean rings, but you can't just ...
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Classification of links made of rigid circles
I am not very familiar with low-dimensional topology and I was wondering if we know the classification of links (in $\mathbb R^3$) that can be isotoped into a position where every link is a rigid (...
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How to discribe the complement of a Seifert surface
We can construct a covering space of a knot complement by cutting along a Seifert surface and glue several copies together. So I want to know how the complement of a Seifert surface looks like.
What I ...
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Is $K_1\# K_2$ isotopic to $K_1\#\overline{K_2}$?
Let $K_1$ and $K_2$ be two oriented knots (with fixed embeddings in $S^3$). Then we have a well-defined oriented connect sum $K_1\#K_2$. We can also take $K_2$ with the opposite orientation to form $...
- 1,366
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The uniqueness of minimal genus Seifert surfaces for knots in $S^3$.
I was reading some materials about knots, some procedures inspired me to ask this question. Given a knot $K$ in $S^3$, one can use Seifert's algorithm to obtain a surface in $S^3$ whose boundary is $K$...
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Proof of the Wirtinger Presentation using Van Kampen Theorem
I have some difficulties understanding a proof of the Wirtinger presentation using the Van Kampen theorem, found in John Stiwell's "Classical Topology and Combinatorial Group Theory".
I ...
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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 ...
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A torus whose middle is tied in an overhand knot; how many holes does it have?
Motivation:
I stumbled upon the following image on Twitter recently:
Description: A see-through torus is depicted in which the "tube" that ordinarily forms its middle is tied in an overhand ...
- 43k
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Rolfsen exercise, torus knots
I am working on exercise 6, p23 in Rolfsen's Knots and Links:
Show that if $G$ is any simple closed curve in $\mathbb{C}-\{0\}$, then within any $\epsilon$-neighborhood of $G$ lies a simple closed ...
- 3
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Existence of normal crossing of knot diagram via Mather's stable mapping theory
Using Mather's theorem on stable mappings/using multijet transversality, we know that the set of immersion with normal crossing $\mathcal M\subseteq C^\infty(S^1,\mathbb R^2)$ is dense and open in the ...
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Determining whether two groups are isomorphic based on Wirtinger presentation
I am working on knot theory and basic algebraic topology. In order to prove that the figure eight knot and the trefoil knot are not isotopic, I have to show that their knot groups (i.e. the ...
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Simplicial approximation of contractible homotopy in "Knots" (by Burde, Zieschang, Heusener)
I have a question about a line in the proof of
3.4 Theorem about Wirtinger presentation (Chapter 3.B, page 34) of the knot
group
$G(K)= \pi_1(\overline{S^3-K})$ of a knot $K \subset S^3$ from "...
- 31
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Typo in Morton Knots notes?
apologies if I'm just missing something obvious. I'm reading the H.R.Morton "Knots and Links" notes with no background, just for fun, and I'm in section 2, where two great circles have been ...
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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 ...
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Does the abelianization of the fundamental group of this knot-complement really $\mathbb{Z}$?
I have a doubt about some edge cases in problem $22.(b)$ in Hatcher's "Algebraic Topology", section $1.2$.
The writer presents a knot:
To begin, we position the knot to lie almost flat on
a ...
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Does the Vassiliev $v_2$ invariant satisfy a relation similar to the Arf invariant?
The Arf knot invariant $\operatorname{Arf}:\{{\rm knots}\}\to\Bbb Z/2\Bbb Z$ satisfies the relation
$$\operatorname{Arf}(K_+) + \operatorname{Arf}(K_-) \equiv \operatorname{lk}(L_1, L_2) \pmod{2}$$
...
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What is the minimal number of edges for a polygonal knot to be nontrivial?
I started learning about knots from Manturov's monograph. Exercise 2.1 says "Show that all polygonal links with less than six edges are trivial". I kind of convince myself why this should ...
2
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Why in $p$-colorability , $p$ must be a prime ? (knot theory)
We say that a knot is $p$-colorable, where $p\ge 3$ is prime, if you can label the strands of the knot using labels from $\{0,1,2,...,p-1\}$, such that at each crossing, $$x+y=2z\pmod p,$$ where $x$ ...
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Nomenclature for links of 2-bouquet homotopic class
I've already understood the basic nomenclature for knot and links, especially links consisting trivial knots (circle). However, things got much complicated when I was working on the case when circles ...
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How does KnotInfo classify knots based on DT Code?
The site KnotInfo can classify many small knots up to mirroring given a Dowker–Thistlethwaite Code of the knot, and can even identify whether a knot is a connected sum of other small knots.
Being very ...
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Why should boundary maps be degree 0 in Khovanov Homology?
In Bar-Natan's On Khovanov's categorification of the Jones polynomial (https://arxiv.org/abs/math/0201043), the claim in section 3.2 when constructing the differential $d_\xi$ is that $d_\xi$ ought to ...
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Existence of a cone neighbourhood in an open disc with a point in the border
I started reading C. P. Rourke and B. J. Sanderson’s “Introduction to Piecewise-Linear Topology” which is recommended by D. Rolfsen in “Knots and Links”, they provide the following definition:
1.1 A ...
2
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Fast determinant of an Alexander matrix
I want to compute the determinant of a polynomial $n \times n$ matrix where each entry is a univariate polynomial of degree at most $1$. I calculated it naively and was quickly reminded of how fast $n!...
3
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1
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79
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Jones polynomial of a knot in terms of its Seifert matrix
It is well known that the Alexander polynomial of a knot can be written in terms of the Seifert matrix of the knot by a simple relationship $$\Delta(t)=\det(V^T-tV),$$ where $t$ is a formal variable ...
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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_+ ...
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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 ...
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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-...
- 156
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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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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 ...
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Describing the homotopy explicitly.
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 ...
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Proving that the cone over a knot is not locally flat
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 ...
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