**5**

votes

**0**answers

89 views

### Ribbon knot presentations

Suppose $K$ is a $n$-knot, $n\geq 2$, which bounds two different ribbon disks $D_1, D_2$. These ribbon disks induce unique ribbon $(n+1)$-knots $K_1, K_2$ respectively. Is it known whether $K_1$ and ...

**12**

votes

**4**answers

830 views

### What properties of knots lead Lord Kelvin to hypothesize that atoms were knots in the ether?

I've often heard that Lord Kelvin was one of the first people to study knot theory, as he hypothesized that atoms were knots in the ether. I assume that he had some compelling evidence for this fact.
...

**2**

votes

**1**answer

114 views

### Is there a criterion for a link complement to have a hyperbolic structure with finite volume

For many links in $S^3$, the link complement can be equipped with a Riemannian structure which is complete, of constant sectional curvature -1, and has finite volume (i.e., a hyperbolic structure with ...

**1**

vote

**1**answer

112 views

### Bitangent locus of torus knots

Anyone know how to compute the bitangent locus of a space curve, e.g. a torus knot (pick whatever parametrization you like)? Specifically, what is the set of normal vectors (in the two-sphere) of ...

**1**

vote

**0**answers

94 views

### Entangled helical knots

Consider a pair of disjoint, congruent helices $H_1$ and $H_2$
passing through one another in the following sense.
(Caveat lector: This question is not of general interest! It is also long.)
$H_1$ is ...

**8**

votes

**3**answers

337 views

### What constant ensures hyperbolicity of Dehn surgery?

I am interested in showing that certain knots having a surgery description are hyperbolic. Unfortunately I have not had time yet to read Thurston's work, so my understanding of this is vague. But from ...

**1**

vote

**2**answers

195 views

### Freely Periodic map of $(S^{3} , K) $ and a fixed loop in the induced isomorphism of $\pi_{1} ( S^{3} \backslash K )$

Let $K$ be a link in $S^{3}$ and $f: S^{3} \rightarrow S^{3} $ a freely periodic map of order $n$ with $f(K) = K$. Let $\psi_{f} : \pi_{1} ( S^{3} \backslash K ) \rightarrow \pi_{1} ( S^{3} ...

**4**

votes

**1**answer

208 views

### The cyclic branched covers of “simple” knots in $S^3$

Is there a convenient place in the literature where the geometric decompositions of cyclic branched covers of $S^3$ branched over "small" knots is recorded?
By small knots, I'm referring to things ...

**7**

votes

**1**answer

170 views

### Is there a combinatorial version of PL ambient isotopy in dimension $>3$?

The classical PL Reidemeister Theorem reads:
Reidemeister Theorem: Two knots in $S^3$ are PL ambient isotopic if and only if any diagram of one can be transformed into a diagram of the other by ...

**6**

votes

**1**answer

167 views

### Knots indistinguishable by HOMFLY

Is there any list (incomplete of course) of knots, that have similar HOMFLY polynomials? I am mainly interested in torus knots.

**4**

votes

**2**answers

181 views

### Getting surgery link from Heegaard splitting

From Lickorish-Wallace theorem, every 3-manifold is an integral surgery on a link in $S^3$. From its proof from Saveliev's book, it seems obvious that if I know the Heegaard splitting of a closed ...

**7**

votes

**2**answers

257 views

### How do I efficiently find a sequence of Reidemeister moves between equivalent link diagrams?

In knot theory, two link diagrams are equivalent if and only if they can be related by performing a finite number of Reidemeister moves. But sometimes it is so confusing that I don't know which type ...

**11**

votes

**1**answer

635 views

### Kontsevich integral : state of the art

The Kontsevich integral is known to be a universal Vassiliev invariant.
It is still an open question whether it is a complete knot invariant, i.e. whether it distinguishes a given knot from all other ...

**15**

votes

**3**answers

597 views

### What is the state of the art for algorithmic knot simplification?

Question: Given a `hard' diagram of a knot, with over a hundred crossings, what is the best algorithm and software tool to simplify it? Will it also simplify virtual knot diagrams, tangle diagrams, ...

**3**

votes

**0**answers

124 views

### More questions about high-dimensional knot invariants

In a question yesterday I asked about the existence of algebraic invariants for embeddings of n-manifolds into n+2-spheres. The answers in the positive dimension all made certain assumptions about ...

**9**

votes

**3**answers

316 views

### Invariants of high-dimensional knots

In the study of knots in three dimensions, it can be shown that the fundamental group together with a specification of a meridian and longitude form a complete invariant for knots. What is known about ...

**1**

vote

**2**answers

289 views

### Commutativity in the Fundamental Group and Knot Theory

Let $M$ be a connected $3$-manifold and let $\alpha$ and $\beta$ be elements in $\pi_1(M)$. Then $\alpha$ and $\beta$ can be represented by two knots $a$ and $b$ in $M$. We may further require that ...

**7**

votes

**3**answers

484 views

### Knots and Dynamics. Recent breakthroughs?

I recently started reading Étienne Ghys slides on knots and dynamics <http://www.umpa.ens-lyon.fr/~ghys/articles/icm.pdf> which seem very interesting. I know this approach to knots and dynamics is ...

**16**

votes

**3**answers

458 views

### Looking for “large knot” examples

This question is about knots and links in the 3-sphere. I want to find an example of a "large" knot or link with some special properties. I'm looking for some fairly specific examples, but I'm also ...

**12**

votes

**3**answers

595 views

### Representing SU(3) with 3 ropes in 3 dimensions

The short question is: how exactly is SU(3) realized with ropes?
The long question: There is this idea that deformations of a configuration of three infinitely long, flexible ropes that cross each ...

**4**

votes

**0**answers

106 views

### Is the normalised Kauffman bracket more powerful than the Kauffman bracket?

The Kauffman bracket polynomial for a knot diagram $D$ is a Laurent polynomial $\langle D \rangle \in \mathbb{Z}[A, A^{-1}]$. Although it is invariant under Reidemeister moves of type II and III, ...

**7**

votes

**2**answers

181 views

### How does Thurston's Orbifold Geometrization imply that knots with meridional rank 2 are 2-bridge?

Problem 1.11 of Kirby's list asks whether every knot that has a knot group
which can be generated by n meridians, but not less than n, is an n-bridge
knot. There is a one-sentence update, saying that ...

**1**

vote

**1**answer

135 views

### Two links with the same signatures but unknown if they are related by Kirby moves

I am wondering if there are links $L_1, L_2$ in the sphere $S^3$ such that:
the signatures of $L_1, L_2$ are known.
we do not know if they are related by Kirby moves.
If so, could you specify the ...

**5**

votes

**1**answer

255 views

### Kauffman's state model for the Alexander polynomial, via representation theory

I've been reading Oleg Viro's paper on "quantum relatives of the Alexander polynomial" (arXiv:math/0204290), which, among other more general things, derives state-sum formulas for the Alexander ...

**6**

votes

**1**answer

235 views

### Piecewise Smooth Knot Theory

In introductory knot theory books, authors usually make a choice of smooth knots or piecewise-linear knots. I often find myself wanting to work in the larger setting of piecewise-smooth knots which ...

**3**

votes

**0**answers

132 views

### Is a generic link diagram semi-adequate?

Each crossing in a link diagram of a link $L$ has an A-resolution and a B-resolution.
Resolving all crossings gives a collection of circles in the plane, connected by dotted lines. A state of a ...

**1**

vote

**2**answers

122 views

### Knot group epimorphism from a prime knot

The knot group of a knot $K$ is the fundamental group of the knot complement $\pi_{1} (S^{3} \backslash K )$ (sometimes denoted $\pi_{1} (K)$ ).
Let $f: \pi_{1} (K) \rightarrow \pi_{1} (L) $ be a ...

**1**

vote

**1**answer

230 views

### The knot group of a prime knot

The fundamental group of a knot $K$ (otherwise known as the knot group) is the fundamental group of the knot complement $S^{3} \backslash K $ in $S^{3} $.
In "Virtual Knots: The State of the Art" ...

**9**

votes

**1**answer

340 views

### What are the homotopy classes of two-component links in $\mathbb{RP}^3$?

This question comes from an unanswered question on Math Stack Exchange.
A two-component link in $\mathbb{RP}^3$ is any embedding $S^1\uplus S^1\to \mathbb{RP}^3$. Two such links are homotopic if ...

**5**

votes

**1**answer

161 views

### Reference for a theorem on crossing changes of links

I've recently stumbled upon a paper of Scharlemann on crossing changes:
link text
In particular I am interested in understanding Theorem 2.2 (page 6):
"Theorem: If links A and B
are related by a ...

**12**

votes

**2**answers

356 views

### Semidirect product decomposition of the Borromean rings group

Let $X=S^3\setminus B$ be the link complement of the Borromean rings.
Then $G=\pi_1(X)$ has a presentation of the form
$$
G = \langle \; a,b,c \mid [a,[b^{-1},c]],\; [b,[c^{-1},a]], \; ...

**2**

votes

**1**answer

134 views

### {0,1} Maslov potentials on Legendrian knots

A Legendrian knot is a curve in $\mathbb{R}^3$ on which $dz - ydx$ vanishes identically. Its projection to the $x,z$ plane is called a front diagram; as we can recover $y = dz/dx$ this determines the ...

**2**

votes

**1**answer

164 views

### Is every quasipositive knot strongly quasipositive?

A link is called quasipositive if it has a special braid diagram, namely a product of conjugates of the positive standard generators of the braid group. If this product only contains words of the form ...

**7**

votes

**3**answers

199 views

### Measures of entangledness of an open curve

Let $\gamma$ be a simple (non-self-intersecting) open curve in $\mathbb{R}^3$.
I am seeking a measure of its degree of "entangledness," some measure that accords
with the intuition one senses with a ...

**2**

votes

**0**answers

93 views

### Why do knot cobordisms result in functoriality with respect to knot homologies so often?

Why do knot cobordisms result in functoriality with respect to knot homologies so often?

**10**

votes

**2**answers

345 views

### Are small knots generic?

A knot in S^3 is small if its complement does not contain a closed incompressible surface. Is it a generic property for knots, meaning that among all knots with less than $n$ crossings, the proportion ...

**4**

votes

**2**answers

230 views

### Can distinct open knots correspond to the same closed knot?

A topological ("closed") knot is an embedding of a circle in $\mathbb{R}^3$. It's possible for a knot to be distinct from the unknot because there are no free ends to move around and untie the knot. ...

**8**

votes

**1**answer

334 views

### When do two positive braids represent the same link?

Let $B_n$ be the braid group on $n$ strands, with the usual generators: $s_1, \ldots, s_{n-1}$ and their inverses, where $s_i$ is a positive half-twist interchanging the strands labelled $i$ and ...

**1**

vote

**2**answers

113 views

### Questions about knot (link) of surface in four dimension

Consider three 2-torus ($S^1*S^1$) living in four space. Can I have links of these objects, which is generalization of links of circles in 3D? If so, how can I judge whether three 2-torus are linked ...

**2**

votes

**2**answers

524 views

### How to compute the Alexander polynomial of general torus knot

Hello,
i am very interested in knot theory, especially in knot groups and knot polynomials. Therefore i am reading the book of Crowell and Fox (Introduction to knot theory). I want to compute some ...

**2**

votes

**1**answer

101 views

### How to visulize surface link in four dimension?

I am now facing a problem with "surface link" in four dimension. I have heard that three 2-torus can be linked in four dimension. And I have created a movie by cutting four dimensional space with ...

**10**

votes

**1**answer

205 views

### Links which HOMFLY homology distinguish but the HOMFLY polynomial does not.

Does anyone know of a pair of different links which the HOMFLY polynomial does not distinguish, but HOMFLY homology does? Or does there exist such a pair of links?
I'm assuming there does exist ...

**3**

votes

**1**answer

181 views

### Satellite knot example

Can someone provide me with an example of a satellite knot with symmetry group which is neither cyclic nor dihedral?

**14**

votes

**2**answers

250 views

### Random rings linked into one component?

Let $S$ be a sphere of unit radius.
Let $C_n$ be a collection of unit-radius circles/rings whose centers
are (uniformly distributed)
random points in $S$, and which are oriented (tilted) randomly ...

**16**

votes

**3**answers

609 views

### Visualize Fourth Homotopy Group of $S^2$

I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in ...

**8**

votes

**3**answers

484 views

### Fibered knot with periodic homological monodromy

It is well-known that there exist pseudo-Anosov automorphisms of surfaces that act trivially on the homology: they form the Torelli group. Similarly there exists pseudo-Anosov automorphisms that act ...

**3**

votes

**2**answers

185 views

### Oriented Reidemeister move R2d by splices and loop adding/erasing?

Consider the move R2d which applies to tangle diagrams, described in the Figure 1.
Question: is it possible to achieve the move R2d by a sequence of splices and loop adding or erasing, i.e. is ...

**9**

votes

**0**answers

289 views

### Is the connected sum of knots an isometry?

Take $X$ as the set of knots in the 3-sphere (i.e. smooth embeddings of $S^1$ in $S^3$ up to smooth isotopy), endowed with the Gordian distance $d$.
For a fixed knot $K$ we can define the map ...

**1**

vote

**1**answer

442 views

### Dehn presentation of a knot group

The knot group is the fundamental group of the knot complement in $S^{3} $. The Dehn presentation of the knot group is a particular group presentation obtained by looking at the regions and crossings ...

**7**

votes

**1**answer

435 views

### Diagrammatic proof of unique prime decomposition of knots

Consider a knot to be a diagram in a plane--- i.e. a drawing of a finite connected planar graph (loops and multiple edges allowed) whose vertices are 4-valent with cyclic ordering for the incident ...