Knot theory is dealing with embedding of curves in manifolds of dimension 3. A knot is a single circle embedded in the affine space of dimension 3 as a smooth curve not crossing itself. Many knot invariants are known and can be used to distinguish knots.

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Connection between cardiac equations and untangling knots?

I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots: Maucher, Fabian, and Paul ...
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1answer
74 views

Looking for examples of large hyperbolic two-generator knots or 3-manifolds

We say a knot $K$ in $S^3$ is small if its exterior contains no closed properly embedded incompressible surfaces and we say $K$ is large otherwise. Does anyone know of an example of a large ...
8
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1answer
195 views

What does the representation category of the knot group know?

Let $K, K'$ be knots in $S^3$, and $T, T'$ the boundaries of their tubular neighborhoods. Recall that by theorems of Waldhausen, and Gordon and Luecke, one knows the following: an isomorphism ...
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3answers
124 views

Knot invariants with skein relation of order 3 or 4

Many of the well-known knot/link invariants $I$, such as the Jones, Alexander, HOMFLY, Conway polynomials, satisfy a quadratic skein relation, i.e. an equation of the form $\alpha I(L_+)+\beta ...
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A three-rope tangle that is equal to its mirror tangle

In the same way that a figure-eight knot is equal (after a suitable rotation) to its mirror knot, I am looking for simple tangles made of three ropes that are equal to their mirror tangle. The ...
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154 views

Are knots determined by their complements within a homotopy class?

Suppose $M$ is a closed 3-manifold and $K_1,K_2$ are two homotopic knots in $M$. That is, they are two embeddings $f_1,f_2\colon S^1 \to M$ such that there exists a homotopy $h\colon S^1 \times [0,1] ...
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3answers
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Is anything known about this braid group quotient?

Let $B_n$ be the braid group on $n$ strands. As is well known, if $\sigma_i$ is the operation of crossing the string in position $i$ over the string in position $i+1$, then the elements ...
8
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535 views

Does anyone know this sequence of polynomials?

A referee on a paper of mine showed me the following recurrence for polynomials $P_{n,k}\in\mathbb Q[q,q^{-1}]$ for $n\geq 0$ and $0\leq k\leq n/2$. \begin{align} P_{0,0}&=1\\ \text{for $n\geq ...
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75 views

Compatibility of spherical and hyperbolic geometry for fibred knots

Hyperbolic knots and links have a lovely peculiarity that you can always find a position for them in $S^3$ making two groups the same, one defined using the spherical geometry of $S^3$, and the other ...
4
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1answer
158 views

Fibered knots vs Heegaard genus

We call a knot $K$ in a 3-manifold $M$ fibered if $M\backslash K$ fibers over $S^1$ with fibers $\Sigma$ and such that $K$ is ambient isotopic to the boundary of the compactified fiber ...
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1answer
197 views

Understanding “Decategorified” symplectic Khovanov homology

In http://arxiv.org/abs/math/0405089 Seidel and Smith constructed a link invariant using Lagrangian Floer theory that was conjectured to be equivalent to Khovanov homology. The equivalence was ...
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1answer
159 views

How many quadratic fields occur as trace fields of hyperbolic knot complements?

I am interested in when the trace field of a knot complement has the form $F(\sqrt{-d})$ for $F\subset\mathbb{R}$ and $d\in F^+$ (squarefree). Does this occur for infinitely many choices of pairs ...
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3answers
244 views

Polynomial invariants for unoriented links

I have seen that usually one finds polynomial invariants for oriented links (for example the Jones polynomial, the Hompfly polynomial). Does anyone know what polynomial invariants exist for ...
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0answers
132 views

Tying knots in $\mathbb{C}$

As far as I know, there is no deep significance to this question, but I've been playing around with it for a bit and it seems interesting: Fix a complex number $c$, and consider the map $J_c: ...
5
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73 views

How long does it take for the action of the braid monoids on Laver tables to become trivial?

Let $A_{n}$ denote the classical Laver table of cardinality $2^{n}$. Let $B_{n}^{*}$ denote the positive (including the identity) braid monoid on $n$ elements generated by ...
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3answers
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What part is left unsolved in the Unknotting problem? (after results of Bar-Natan, Khovanov, Kronheimer and Mrowka)

Kronheimer and Mrowka showed that the Khovanov homology detects the unknot. Bar-Natan showed a program to compute the Khovanov homology fast: there was no rigorous complexity analysis of the ...
3
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57 views

Linking circles inside an immersed surface

(Migrated from Math Stack Exchange) A smooth embedding $f : D \to \mathbb{R}^3$ can be isotoped to a canonical inclusion $D \hookrightarrow \mathbb{R}^3$. (This is part of a proof that only the ...
4
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1answer
653 views

Knot database including text names

Knots such as the 3_1 knot and the 4_1 knot are often referred to as the trefoil and figure-eight knots respectively. There are more obscure names for some of the later ones in the knot tables, for ...
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2answers
190 views

What can one do with randomly generated parameterized knots in 3 space?

I have a way of generating random parameterized maps from $S^1 \to \mathbb{R}^3$. This method can create very simple knots, such as ellipses, but can also create knots with more crossings than I can ...
3
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1answer
84 views

Can the ribbon category of f.d. reps of $\mathcal{U}_q(\mathfrak{sl}(2))$ be modified so the twist is trivial on the vector representation?

Consider the ribbon category of finite-dimensional representations of $\mathcal{U}_q(\mathfrak{sl}(2))$, with twist $\theta$. If $V$ is the vector representation, then $\theta_V$ is multiplication by ...
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2answers
613 views

What is the mathematical significance of the IHES logo?

The logo of the IHES http://www.ihes.fr/jsp/site/Portal.jsp (upper left) is lovely, but what exactly does represent mathematically? (There's a slightly larger version at ...
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Are there moves between Reidemeister moves?

Background Knots are typically written in 2 dimensions as a loop in the plane with normal crossings. One then asks when two such diagrams describe the same knot. Two diagrams describe the same knot ...
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1answer
349 views

Is a “knot knot” or “double knot” a thing in knot theory?

I apologize in advance for my rudimentary knowledge of knot theory, but I've been trying to find out about the significance (if any) of taking a knot (particularly a torus knot), cutting it, and ...
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658 views

Classification of knots by geometrization theorem

I read this interview with Ian Agol, where he says: "...I learned that Thurston’s geometrization theorem allowed a complete and practical classification of knots." My question is: How does ...
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3answers
500 views

Algorithm for detecting ribbon or slice links?

A link in $S^3$ is said to be slice if it bounds a collection of flat disks into the $4$-ball. Here "flat," means that there is a (locally) trivial normal bundle. This condition can be strengthened to ...
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1answer
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Ascending surfaces in the 4-ball

Let the standard symplectic structure on $B^4$ (viewed in $\mathbb{R}^4$ or $\mathbb{C}^2$) be given by $\omega=(1/2) d \eta$, for \begin{align*} \eta &:= x_1 \, dy_1 - y_1 \, dx_1 + x_2 \, ...
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9answers
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Applications of knot theory

An answer of André Henriques' inspired the following closely related CW question. Parts of the following is extracted from his answer and my comments. I regularly teach a knot theory class. ...
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1answer
180 views

On Knot Equivalence problem statement

How is the knot equivalence problem represented? By this I mean I am looking for an analogy that compares with graph equivalence. For graph equivalence, we have two graphs $G_1$ and $G_2$ with ...
11
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4answers
739 views

Distance between two knots

Are there some well-studied functions defining natural distance measures between two knots? One can imagine a function that counts, say, the minimum number of moves, each of which passes one strand of ...
7
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4answers
411 views

hyperbolic structure on Figure–8 knot complement

I was trying to understand the proof of the fact that there is a hyperbolic structure on Figure–8 knot complement initially from Thurston's notes and then from some online notes; but unfortunately I ...
7
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2answers
195 views

Can every large point set be connected to a given knot?

Let $K$ be a given knot, and $P$ a set of points in $\mathbb{R}^3$ in general position, general position in the sense that no three points are collinear and no four coplanar. Define the point-set ...
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Knot invariants which arise as the solution of differential equations along the knot

Do there exist knot invariants which can be computed by solving some kind of differential equation along the knot?
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90 views

Unnormalized Kauffman homology of the unknot

Is the unnormalized Kauffman homology of the unknot known? The Poincare polynomial of HOMFLY homology of the unknot is known as $$\frac{1+at}{1-q}.$$ Is the Poincare polynomials of Kauffman homology ...
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1answer
95 views

Cancellation of 2-component links

Consider a two-component tame link in 3-space, consisting of an arc from $(-1,1,0)$ to $(1,1,0)$ and an arc from $(-1,-1,0)$ to $(1,-1,0)$, confined to the slab $-1 \leq x \leq 1$. Call such a link ...
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3answers
456 views

Classification of tangles?

Has anybody done any work on making a classification of low-complexity tangles, analogous to the work for knots and links? I expect most of the small ones to be rational, and those that aren't ...
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1answer
203 views

Can one twist fibred knots and still get fibred knots?

Suppose we have a fibred knot $K$ with a fiber surface $F$ and let $c$ be an unknot disjoint from $F$ (but not homotopically trivial in the complement of $F$). Is it possible that every twist along ...
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1answer
55 views

Calculating the Upper Bound on the Sphere Radius of Knotted Channel Surfaces

This question is motivated by trying to determine the upper bound on the thickness of a rope of fixed length (w.l.o.g. $2\pi$), with which a knot of given topology can be realized under the further ...
21
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1answer
410 views

Why, in terms of quantum groups, does the knot determinant appear as an evaluation of both the Jones and Alexander polynomials?

The Jones polynomial can be computed from the representation theory of $\mathcal{U}_q(\mathfrak{sl}(2))$. The Alexander polynomial has an analogous description in terms of the representation theory of ...
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1answer
182 views

Physical strength of a link [closed]

Assuming that we construct a link/chain using a collection of knots. Is there a way to measure the physical strength of this chain?
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256 views

Fibered example of topologically slice knots

Is there any known example of fibered knot which is topologically slice but not (expected to be) smoothly slice?
8
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1answer
204 views

Questions on poincare homology spheres and branched covers

I have two questions: Question 1. Suppose that $K$ is a knot in $S^3$. Let $\Sigma(K)$ be the double branched cover of $S^3$ branched along $K$. If $\Sigma(K)=\#_{i=1}^n\Sigma(2,3,5)$, then ...
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Concordance and homology cobordism

If two knots $K_1$ and $K_2$ in $S^3$ are smoothly concordant, then for any rational number $r$, the $r$-surgeries $S^3_r(K_1)$ and $S^3_r(K_2)$ are homology cobordant. Is the converse true? What if ...
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1answer
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Understanding Penrose diagrammatical notation

I arrived to Penrose's paper Applications of negative dimensional Tensors after reading some bits of Baez's Prehistory (link) and the first two chapters of Turaev's Quantum invariants of knots and ...
6
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1answer
251 views

Satellite knot example

Can someone provide me with an example of a satellite knot with symmetry group which is neither cyclic nor dihedral?
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Is Rasmussen's s-invariant of a knot an invariant of a 4-manifold?

Let $K$ be a knot in the 3-sphere $S^3$. Here we denote by $s(K)$ Rasmussen's s-invariant for $K$, and by $X_{K}(n)$ the 4-manifold obtained from the standard 4-ball $B^4$ by attaching a ...
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1answer
502 views

A fun game related to knot theory

I recently learned the following rather fun game: a group of people is standing up roughly in circle, facing each other. Then participants randomly join hands, in such a way that nobody holds its own ...
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Surgery along an arc connecting the components of a $2$-component link gives the unknot

Math Overflow seems to have a dearth of low dimensional topology, but this seems like an interesting question. Let $L$ be a $2$-component link in $S^3$. Suppose that there is a framed arc joining the ...
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1answer
90 views

Essential surfaces in the Exterior of Montesinos knots

Hatcher and Oertel computed the boundary slopes of essential surfaces of Montesinos knots in this paper. But they do not consider surfaces that do not intersect the boundary of the exterior. An ...
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Metrics on the 3-sphere with knotted geodesics

According to answers to this question every metrics on $S^3$ admits a simple closed geodesic. Given a knot (or link) $K$, it's also quite simple to build a metric on $S^3$ such that $K$ is a geodesic ...
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Are Fourier series of length 2 'asymmetric enough' to generate all crossing patterns? - A reformulation of the Fourier-(1,1,2) knot question

Given $N$ pairs of distinct real numbers $t_i, t'_i \in [0,1]$, $i = 1,\ldots,N$, we ask if there is a function $f(x) = \cos(2\pi mx+\alpha) + \gamma\cdot \cos(2\pi nx+\beta)$, with $m, n \in ...