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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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
534 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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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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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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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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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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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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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 ...
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4answers
707 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
380 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
190 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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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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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
85 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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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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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
199 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
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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
388 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
181 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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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?
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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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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 ...
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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
440 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
85 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 ...
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What is the easiest way to compute Ozsváth-Szabó tau invariant of a knot?

Suppose that we have a knot $K$ with 40 crossings which is not a cable knot or an alternating knot. Then, what is the easiest way to compute Ozsváth-Szabó's invariant $\tau(K)$? Are there any ...
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Distiguishing mutant knots

Can an invariant from a quantum Lie algebra ever distinguish mutant knots? (Maybe if it is "chiral"...whatever that means :-) (Note that Kauffman abstract tensors/skein equations CAN distinguish ...
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Framed singular knots

I've recently run across what one might (and I suspect people probably do) call framed singular knots, or maybe singular ribbon knots. Regardless of the name, what I mean is the following: Let $D$ be ...
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Union of knots and its Alexander polynomial

In 1957 Kinoshita and Terasaka in their article "On unions of knots" generalized the operation of connected sum; it's called $union$. I have several questions : 1.) If the Alexander polynomial of a ...
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Codimension one embeddings

For smooth knots in $S^3$ "isotopy" and "ambient isotopy" are equivalent (although this is not true in the topological category). I guess that therefore also for tori in $S^3$ "isotopy" and "ambient ...
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Are there any very hard unknots?

Some years ago I took a long piece of string, tied it into a loop, and tried to twist it up into a tangle that I would find hard to untangle. No matter what I did, I could never cause the later me any ...
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Introductory article of knot Heegaard Floer Homology

I am looking for some article that gives an introduction to Heegaard Floer homology of knot. I heard that it is very useful to determine the unknotting number of a knot, but I couldn't find any ...
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3answers
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How to show whether a given knot and its mirror image are the same or not?

The title says it all: How can I show that a knot $K$ is distinct from its mirror image? May be I have to try different knot invariants. Not sure, I am new in this area.
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1answer
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A basic question of Khovanov-Rozansky Homology

thank you for spending time on the following question. In [1] Khovanov and Rozansky categorifized $sl_n$ version of HOMFLY Polynomial, in page 11, they mention that what they defined in [1] is ...
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How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, … algebras?

There is a theorem (I believe by Ocneanu) that the Markov trace on the tower of Temperley-Lieb algebras is (essentially) unique. What about just traces on separate algebras? That is, take one of ...
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Equivalence (or not) of two Artin/Fox wild arcs

The repeating patterns in the wikipedia articles on wild arcs and wild knots seem to me to be not continuously deformable to each other. Is this true? For clarity, here is my diagram of the repeating ...
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1answer
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Does $H_*(A^-_0(K))=\mathbb{F}[U]$ imply that $K$ is an L-space knot?

Let $K$ be a knot in the three-sphere. Let $A_s^-(K)$ be the Alexander filtrations of the knot Floer complex $CFK^{\infty}$. Would $A_0^-(K)$ has homology $\mathbb{F}[U]$ imply that K is an L-space ...
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Kinematics of rolling knots

It is well known that there are trefoil knots without tritangent planes, and with 3d printers one can print these beautiful objects and make them roll on planes. (An ...
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Doubly slice knots and an embedding of 4-manifold

It is well-known that the existence of topologically slice knot which is not smoothly slice implies the existence of exotic $\mathbb{R}^4$. For example, see the answer of K. Davis. To prove the above ...
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Are Markov traces matrix traces?

When starting this question I was very hesitant - literature on the subject is vast and I thought most likely the answer is already there somewhere. Then when the list "Questions that may already ...
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Human Knot game [duplicate]

In the popular game "Human Knot", a group of people stands in a circle and each person grabs another person's hands at random (one with the left hand and one with the right hand). The goal is to ...
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Kernel of “Hat to Plus” in Heegaard Floer Homology

Given a 3-manifold $M$, there is a map of Heegaard Floer groups $$f:\widehat{HF}(M) \to HF^+(M)$$ induced by the inclusion $$\widehat{CF}(M) \to CF^+(M)$$ of the respective chain complexes. Given a ...