Questions tagged [knot-theory]

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

Filter by
Sorted by
Tagged with
-1 votes
0 answers
22 views

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, ...
user avatar
1 vote
0 answers
9 views

Split link properties

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 ...
user avatar
0 votes
0 answers
55 views

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 ...
user avatar
0 votes
0 answers
21 views

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 ...
user avatar
2 votes
0 answers
23 views

Obstruction Theory in Seifert Surfaces

Background: This is from Livingston unplished notes in knot concordance: Proposition 1.7.1 claims: Let $K$ be a knot in $S^3$. Then every Seifert surface $F$ for $K$ has a function $f:S^3-\nu(K)\to S^...
user avatar
0 votes
0 answers
25 views

Is the pair of pants (3 punctured sphere) surface an incompressible surface?

Is the pair of pants (3 punctured sphere) surface an incompressible surface? My intuition says no since a simple closed curve around a boundary component will not bound a disk on the surface.
user avatar
2 votes
0 answers
12 views

Can a rational Seifert surface have n boundary components?

I was wondering if it is possible for a knot to have a rational Seifert surface with 3 boundary components (pair of pants) and n many in general.What would such a knot look like?
user avatar
0 votes
1 answer
32 views

Intersection of Satellite, Torus, and Hyperbolic Knots

Thurston famously proved every knot that is neither a satellite nor a torus knot is hyperbolic. I was thinking about the definition of satellite knots, in particular which cases we should eliminate to ...
user avatar
0 votes
2 answers
43 views

Trying to explain the unknotting number of $7_5$

I am trying to proof that the unknotting number of prime knot $7_5$ is $2$. For this purpose, I am studying the minimal number of crossings that need to be changed in the knot in order to get a ...
user avatar
  • 11
6 votes
0 answers
133 views

Regarding linking number of oriented knots

Let $K,L\subset R^3$ be oriented knots. Assume that $S\subset R^3$ is a two-dimensional compact connected oriented sub-manifold, $\partial S=K$, and the orientation given on $K$ merges with the ...
user avatar
  • 702
1 vote
2 answers
54 views

how did they come up with polynomials as knot invariant

I understand how to calculate a Jones polynomial for a given knot, but I am not sure why would one search for a polynomial for invariants. How did he come up with these calculation rules?
user avatar
  • 144
0 votes
1 answer
17 views

Prove that the knots 4_1 and 5_2 are not equivalent by showing one is p-colorable and the other is not for some prime p.

I've been learning about knot theory and I am a little confused how to prove the above statement. I know that the 4_1 knot (or the figure-eight knot) is not 3-colorable. However, it is 5-colorable. So ...
user avatar
3 votes
1 answer
53 views

Poincare dual of the Alexander dual of the fundamental class of a knot is given by a Seifert surface - proof

Let $K\subset S^3$ be an oriented knot and let $F:\overline{B^2}\times K\rightarrow S^3$ be a thickening with self linking number $0$. I will denote $F(B^2\times K)$ by $(B^2\times K)$ for simplicity. ...
user avatar
1 vote
1 answer
30 views

fundamental class of the boundary of a knot complement

Let $K$ be an oriented knot and let $F:\overline{B^2}\times K\rightarrow S^3$ be a thickening (with self linking number $0$; we do not explicitly need this as we work only with meridians of $K$, but ...
user avatar
2 votes
1 answer
60 views

Riddle: how did Sossinsky code this left trefoil knot?

In the notes at the end of his book Knots, Mathematics with a Twist (2002), Sossinsky leaves a small riddle: how can a knot be recognized by a computer? He gives the example of the left trefoil knot ...
user avatar
2 votes
1 answer
39 views

Isn't a continuous and injective application $f:S^1\to \mathbf{R}^3$ sufficient to define a knot?

Definition: A knot can be defined as an embedding $f:S^1 \to \mathbf{R}^3$, i.e. as a continuous, injective map from $S^1$ to $\mathbf{R^3}$ which realizes a homeomorphism on its image (Porter - Knots ...
user avatar
1 vote
0 answers
30 views

Seifert surface with components

I was wondering what the knot of a Seifert surface with 3 or more boundary components would be. Doesn’t every Seifert surface have one boundary component by definition ?
user avatar
1 vote
1 answer
47 views

Why is the number of components in the torus link $T_{p,q}$ equal to $\gcd(p,q)$?

I was not able to prove this by myself and also have not found any proof online. Since it is often stated as a fact, I assume it should not be a difficult statement to prove. The definition of $T_{p,q}...
user avatar
  • 11
1 vote
0 answers
31 views

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 ...
user avatar
3 votes
1 answer
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 "...
user avatar
1 vote
0 answers
28 views

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 ...
user avatar
0 votes
1 answer
23 views

Summands of a free product with amalgamation

I'm currently reading Freedman's paper on the Mobius energy of knots. In the proof of Theorem 4.3, he constructs a cylindrical covering $N$ of a tame knot $\gamma_K$, contained a tubular neighborhood $...
user avatar
  • 1,255
0 votes
0 answers
36 views

Why is the determinant of the 5-2 knot not a multiple of 5?

A knot $K$ is $p$-colorable if and only if the det($K$) is divisible by $p$. So why is the |det($5_2$)| $=7$, and not a multiple of 5, especially when there are only 5 arcs in the knot?
user avatar
  • 1
1 vote
0 answers
35 views

If two knots are tricolourable does this imply that they are 3 equivalent?

In reading the knot book there was a question regarding that 3 moves preserve tricolouration and pointing to if a knot is tricolourable and 3 equivalent to another knot then the other knot is also ...
user avatar
1 vote
1 answer
48 views

Construction of infinite cyclic cover of knot exterior using seifert surface

To construct infinite cyclic cover of knot exterior using seifert surface following the Lickorish's textbook, I think that we must choose a tubular neighborhood $T$ and a Seifert surface $F$ of the ...
user avatar
  • 121
5 votes
1 answer
102 views

What is the group presentation for the two-twist spun trefoil?

The sources I'm looking at are giving me conflicting information. One paper gives the presentation $$\langle x,y|xyx=yxy, x^2y=yx^2\rangle,$$ while another paper asserts that Example 12 from Fox's A ...
user avatar
1 vote
0 answers
35 views

A null-homologous knot in the solid torus

The following picture is from Saveliev's book Lectures on Topology of 3-manifolds, page 130: He says that this dark-black knot in the solid torus $S^1 \times D^2$ is homologous to $S^1 \times \{ 0\} \...
user avatar
1 vote
1 answer
49 views

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 ...
user avatar
  • 59
2 votes
0 answers
67 views

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 ...
user avatar
3 votes
1 answer
72 views

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 ...
user avatar
0 votes
1 answer
53 views

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?
user avatar
3 votes
0 answers
70 views

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 $H_1(M)$, ...
user avatar
0 votes
0 answers
26 views

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 $n$-...
user avatar
  • 66
1 vote
0 answers
30 views

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 ...
user avatar
2 votes
1 answer
29 views

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 ...
user avatar
1 vote
1 answer
39 views

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 ...
user avatar
  • 580
6 votes
1 answer
53 views

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 ...
user avatar
3 votes
1 answer
64 views

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 $\cdot$ ...
user avatar
1 vote
0 answers
42 views

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 $2$-...
user avatar
2 votes
1 answer
82 views

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 ...
user avatar
0 votes
0 answers
26 views

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 ...
user avatar
1 vote
0 answers
40 views

Genus zero links

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 ...
user avatar
  • 149
3 votes
1 answer
71 views

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 ...
user avatar
  • 467
2 votes
2 answers
41 views

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 ...
user avatar
  • 580
2 votes
0 answers
35 views

Monodromy representation of branched covers

From Hilden-Montesinos theorem, we know that every $3$-Manifold can be constructed as an irregular dihedral 3-fold cover of $S^3$ i.e. there is a represented Knot $K$ such that $ \omega: \pi_1(S^3 - K)...
user avatar
1 vote
1 answer
61 views

What is the bilinear intersection form of real homology of 4-manifold?

I am currently reading Invariants of 3-manifolds via link polynomials and quantum groups by N. Reshetikhin and V.G. Turaev. In section 3.2, given a framed link $L$, we can get a $4$-manifold $D_L$ by ...
user avatar
3 votes
1 answer
138 views

Why is $A - A^T$ unimodular for any Seifert-matrix $A$ of some knot $K$?

I am studying Seifert matrices. Right now I am working with those matrices from a linear algebra perspective, but I will need to understand the knot theory behind it as well. I am using the fact that $...
user avatar
0 votes
0 answers
13 views

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 ...
user avatar
  • 1
3 votes
1 answer
58 views

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\...
user avatar
  • 2,342
1 vote
0 answers
44 views

Reference request for proof of Tait conjectures

Tait conjectures are three or four statements as old as knot theory itself: Any connected, reduced (without isthmus) alternating diagram of a link has minimal number of crossings. Two reduced ...
user avatar

1
2 3 4 5
22