Questions tagged [knot-theory]

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

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knot theory and python

I am working on a school project involving the application of knot theory to the ADN during its replication. I am required to write a python program in a relation with the subject. I am really stuck, ...
1 vote
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I am stucked, to show the following statement: Let $L = L_1 \cup L_2$ be a split link. Show that $det(L)=0$ but $\sigma(L)= \sigma(L_1) + \sigma (L_2)$. Where $det$ means the determinant of the link ...
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The figure eight knot complement in $S^3$.

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise ...
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what are vassiliev invariants and how are they related to Chern-Simmons?

Possibly this question is very open for this platform and a little diffuse but I am a physicist and my knowledge in tides is not that high but I do not understand graphically what the vassiliev ...
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1 vote
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algebraic intersection number and Poincare duality

I am trying to understand the algebraic intersection number in terms of Poincare dual and the cup product. This is: Let $M$ be a compact oriented $m$-dimensional smooth manifold together with a ...
131 views

Regular pentagon folding a strip

For young students it is an interesting surprise to discover that a knot tied in a strip of paper is a regular pentagon. I'm interested to find a simple, but rigorous, geometrical proof of this "...
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1 vote
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Proving the inequality in the Fáry-Milnor Theorem is strict

The Fáry-Milnor Theorem, as stated in Kristopher Tapp's Differential Geometry of Curves and Surfaces, states that, for a unit-speed simple closed (Tapp uses the convention that only regular curves are ...
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1 vote
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Two knots $K$ and $K^\prime$ are equivalent if and only if their projections $P(K)$ and $P(K^\prime )$ are equivalent [closed]

I have been trying to find a proof for this theorem online but can't seem to find one anywhere, and am unsure where to start in terms of trying to write my own proof. I think what is mostly confusing ...
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Ambient isotopic is equivalent to orientation-preserving homeomorphism

I'm currently studying Knot Theory from Gerhard Burde - Heiner Zieschang "Knots". I'm stuck with the proof of $(1) \Rightarrow 2$ the following: The question was asked previously here and ...
• 5,036
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Relationship between HOMFLY and Alexander-Conway polynomials

Using $L^*$ to denote the mirror image of a $\mu$-component link $L$, the HOMFLY polynomial satisfies $P_{L^*}(l,m)=P_L(l^{-1},m)$ while the Alexander-Conway polynomial (i.e. a symmetric ...
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What should I learn to understand Piccirillo's solution of Conway Knot?

I'm very curious to learn how Piccirilo proved that the Conway Knot is not slice. What should I study to understand her paper in details?
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Homology of knot complement in an arbitrary $3$-manifold

Let $K$ be an oriented knot in an oriented $3$-manifold $M$. Let $\nu(K) \approx S^1 \times D^2$ denote the tubular neighborhood of $K$ in $M$. If $K$ is null-homologous, i.e., $[K]=0$ in $H_1(M)$, ...
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What is the locus of all shape parameters of ideal tetrahedra which share the same volume?

I'm taking a class in hyperbolic knot theory out of Jessica Purcell's book, and I was curious about some volumes and classifications of ideal tetrahedra in $\mathbb{H}^3$, with the upper half $n$-...
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1 vote
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Why is Conway notation relevant, instead of braid notation (for knots)

In the early XXth century, Alexander showed that every knot is the closure of a braid, and Markov gave necessary and sufficient conditions for two braids to have the same closure. My question is: why ...
• 128
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Non-orientable genus of Unknot and it's uniqueness

In the book "Topology Now!" by Robert Messer one of the practice problem suggests, " One could define the nonorientable genus of a knot to be zero for the trivial knot, and for any ...
1 vote
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Explicit form of element in a link group

I have a link which is union of knots $K_1\cup\ldots\cup K_n.$ I do know how to find link group $\pi_1(\mathbb{R^3}-K_1\cup\ldots\cup K_{n-1})$, for example, using Wirtinger presentation. What I want ...
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Seeking the longer paper for which John Conway's "An enumeration of knots and links" is "an abbreviated form"

I'm reading the paper "An enumeration of knots and links, and some of their algebraic properties" by John Conway. The first sentence of the first section reads This paper is an abbreviated ...
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change of linking number along Seifert surface is given by algebraic intersection number

Let $J,K,L$ be knots in $S^3$ and $F$ be some surface such that $\partial F = K\cup J$. I want to prove that $$\text{lk}(L,K)=\text{lk}(L,J)+F\cdot L,$$ where lk is the linking number and $\cdot$ ...
1 vote
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Rolfsen: Computing $\pi_1$ from diagrams for $3$- and $4$-manifolds

We can think $$\partial (D^3 \times S^1) = S^2 \times S^1 = D^2 \times S^1 \cup S^1 \times D^2,$$ the latter union comes from the identity map along their same boundaries. So we can attach a $2$-...
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Seifert surfaces of knots

Let $K$ be a null-homologous knot in a $3$-manifold $M$, i.e., $[K] = 0$ in $H_1(M)$. In this post, it was elegantly shown that there is a Seifert surface $F \subset M$ such that $\partial F =K$. I ...
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Opposite orientation on knot's Seifert surface bands

I am currently studying Seifert surfaces and the ability to deform them into disk-band surfaces, such that the bands leave and reattach to the disk with an even number of half-twists in them. An ...
1 vote
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We know that every link in $S^3$ bounds a Seifert surface, and the genus of a link is defined as the minimal genus among the Seifert surfaces it bounds. It is easy to see that the only genus zero knot ...
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Why can Rolfsen use Van Kampen? (Knots and Links)

I am having some difficulty with Rolfsen's derivation of the Wirtinger presentation of a knot in "Knots and Links" (pages 56 to 60). The basic setup is illustrated below. The proof begins ...
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How to prove that braid index of a specific knot is at least 3?

I have a specific knot $K$ (it seems that it is $6_3$) and want to find its braid index. I managed to construct a braid with 3 strings whose closure is $K$, however I do not know whether 2 strings ...
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Suppose $α$ is a braid word, $α ∈$ braid group $B_n$, the exponent sum of $α$ is $e$, and the closure of $α$ is a knot, is $e+n$ odd?

When I calculate the value of Jones polynomial of a knot $K$ at $\exp(2\pi i/3)$ let $m=e+n-1$, I get $$V_K(2\pi i/3)=(-1)^m$$ but in the paper of Jones, $V_K(2\pi i/3)=1$, so I wanted to know suppose ...
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Action of ambient isotopies on the homology of an embedded $n$-torus in $\mathbb{R}^{n+1}$ and $S^{n+1}$ that bring the $n$-torus back to itself
Let $T^n$ be the $n$-torus embedded in $\mathbb{R}^{n+1}$. Consider the ambient isotopies that bring $T^n$ back to itself. These isotopies have an action on $H_1(T^n)$ described by some subgroup \$G_n\...