**9**

votes

**3**answers

375 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

303 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

654 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

613 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

644 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

116 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

216 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

141 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

345 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

313 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

165 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

152 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

341 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

363 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

168 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

459 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]], \; [c,[a^{-1},b]...

**2**

votes

**1**answer

165 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

286 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

208 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

97 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

408 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

252 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

548 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 $i+1$....

**1**

vote

**2**answers

124 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

751 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

112 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

265 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 ...

**6**

votes

**1**answer

255 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

270 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

655 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 $\pi_3(S^2)...

**9**

votes

**3**answers

663 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

228 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

304 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

765 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

525 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 ...

**5**

votes

**1**answer

220 views

### Locally flat non-smooth discs

There are many knots (e.g. the $P(-3,5,7)$-pretzel knot) that are topologically, but not smoothly slice; "topologically" slice means that there is a locally flat embedding of a disc into the 4-space, ...

**10**

votes

**2**answers

512 views

### Genus one fibered links

It is well-known that the only genus one fibered knots are the trefoil and the figure-eight. On the other hand, there exist infinitely many fibered links for any fixed higher genus.
My question is ...

**1**

vote

**0**answers

138 views

### Categorification of WRT invariants of integral homology spheres

First, I would like to know how many definitions are there for categorification of WRT invariants. In addition, I wonder if the categorified version of WRT invariants have been explicitly computed for ...

**1**

vote

**0**answers

102 views

### Can all knot polynomials derived from Skein relations be categorified?

Can all knot polynomials derived from Skein relations be categorified?

**5**

votes

**1**answer

209 views

### Seifert genus of the lift of a knot in its cyclic branched covers.

I was wondering if there are any known examples of knots $K$ in $S^3$ with Seifert genus $g$ so that the lift of $K$ sitting inside its $n$-fold cyclic branched cover bounds an embedded surface of ...

**2**

votes

**0**answers

127 views

### Are isolated knots in tangles the same as ordinary knots?

Suppose you have an arc in a ball with ends fixed on a boundary. In other words, we have an isolated knot on one of the arcs of a tangle. If we glue the ends we get an ordinary knot and clearly ...

**8**

votes

**2**answers

516 views

### Khovanov-Rozansky homology and spectral sequences

In arXiv:math/0607544 (following conjectures in arXiv:math/0505662), Rasmussen constructs a family of spectral sequences (the "d_N differentials"), starting at the HOMFLY homology of a knot, and ...

**3**

votes

**1**answer

209 views

### One question about the quandle

Given a finite quandle $Q$, for any knot $K$ one can associate an invariant, i.e. the number of proper colorings $p(K)$. Let us consider the inverse $K^{-1}$ and mirror image $K'$ of $K$. My queston ...

**8**

votes

**1**answer

255 views

### Existence of different knots in $RP^3$ having the equivalent liftings in $S^3$

I'm looking for the answer to following question. Do exist different knots in $RP^3$ which have equivalent liftings in $S^3$ under covering $p:S^3\rightarrow RP^3$?

**5**

votes

**1**answer

281 views

### Differences between various categories of surface embeddings in 4-space

This is a very naive question, but I'm trying to understand the difference between the various categories when it comes to embedding surfaces in 4-dimensional manifolds. The situation I'd really like ...

**3**

votes

**1**answer

125 views

### Some questions about ideal knots

The ropelength of a knot curve $C$ is defined as the ratio $L(C) = \lambda(C)/ \tau(C)$, where $\lambda(C)$ is the length of $C$ and $\tau(C)$ is the thickness of the knot defined by $C$ [from ...

**6**

votes

**2**answers

315 views

### Vassilliev invariants of knots and their cables

The following is perhaps a standard question, but i could not find a plain enough answer
by simply searching online.
Q: Given a knot $K$ and its $(p,q)$-cable $K_{p,q}$ what is a relation
between ...

**6**

votes

**1**answer

473 views

### integer surgeries on knots

I have constructed a list of surgery coefficients which yield spherical space forms. For instance, there are two knots with different Alexander polynomials on which 29-surgery will give a small ...

**10**

votes

**3**answers

420 views

### Is a knotted trivalent graph determined by its set of unzips?

A knotted trivalent graph (KTG) is a framed embedding of a trivalent graph in $\mathbb{R}^3$, modulo ambient isotopy. I believe they were first considered (in the quantum topological context, at least)...

**7**

votes

**2**answers

383 views

### Original proof of the existence of Seifert surfaces

I read on Wikipedia that Frankl and Prontrjagin were the first to prove that a link $\mathbb{R}^3$ bounds a surface. A few years later Seifert published a proof using the "Seifert algorithm" which ...