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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Measuring the complexity of a knot by minimum number of simplices to tile the complement

This is essentially a duplicate of: Lower bound on number of tetrahedra needed to triangulate a knot complement Suppose a knot $K\subseteq\mathbb S^3$ is such that the complement $\mathbb ...
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Growth of knots possible with rope of length L

What is the asymptotics (in L) for the number of topologically different knots possible using a perfectly flexible, non-selfintersecting rope of length L and radius 1? (With ends glued together after ...
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Why “Categorify”? Relating to link/knot homologies…

Hey Everyone! So I am new blood in the topic of Khovanov Homology and related topics. According to my basic reading the idea is to get the Jones polynomial as the Euler Characteristic of a certain ...
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Self-tightening knot

Is there a way, for some finite L>1, to tie two pieces of rope together, such that any finite force is not enough to pull them apart? The type of rope I have in mind is something like cylindrical ...
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Knots that are not knots [closed]

1) Classic Knotting problem: Classify embeddings of circle into 3D Euclidean space up to isotopy. http://en.wikipedia.org/wiki/Knot_theory 2) General topological knotting problem: Classify embeddings ...
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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 $0\leq k\leq n/2$. ...
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329 views

Connected Sum Decomposition of a Knot

Given a composite knot, is it possible to decompose it in prime knots by an algorithm that runs in polynomial time?
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357 views

Is there a known method for finding the minimum bridge index of a knot?

It is easy to establish an upper bound $n$ for the bridge index of a knot by producing a diagram with the knot in $n$-bridge position. Is there a known method to produce a reasonable lower bound ...
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849 views

Is there a periodic table for knots?

When I see knot tables, I have two feeling: ah, it's beautiful, and... painful. I don't see how knots are ordered in the knot table, the way to go from one knot of a certain crossing number to ...
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215 views

a special type of 2 component link complement

It is well know that a 2 component link complement may doesn't detect the link type. My question is whether the following type of 2 component links detect their link types? Such a link is composed ...
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234 views

Knots that turn around an axis [closed]

Take a thick cord (the alim cord of your laptop or your mouse cord for example) and wrap it around your hand (or finger) turning always in the same direction but possibly knotting it. Then try to ...
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608 views

Does there exist infinitely many prime knots?

I'm not a topologist and I just saw the definition of prime knot a while ago. Today I'm somewhat supprised to realize that I don't even know if there are infinitely many prime knots? If this ...
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Knot polynomials of non-crystallographic Coxeter groups?

I learnt that the Coxeter groups have a few members more than the classic simple Lie groups: $H_3, H_4$ and $I_2(p)$. Is there a Reshetikhin-Turaev invariant for those, too? If not, where does the ...
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268 views

First cohomology of the space of long knots in R^4

Let's consider the space of long knots in $\mathbb R^n, n>3$. I know that there are many results (Vassiliev, Turchin, Sinha, Kontsevich) about different expressions of cohomology of this space. I ...
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474 views

Traces on Hecke algebras and the Jones polynomial

In his famous paper "Hecke algebra representations of braid groups and link polynomials," (Annals 1987), Jones uses a compatible family of traces $tr_z$ on the Iwahori-Hecke algebras $H(q,n)$ of type ...
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724 views

Fox differential calculus and the Alexander invariant of a link

I am teaching a course in knot theory, and I would like to describe the presentation of the Alexander module of a link obtained via Fox differential calculus. In doing this, I should prove the ...
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Is there a “knot theory” for graphs?

I think knot theory has been studied for quite a while (like a century or so), so I'm just wondering whether there is a "knot theory" for graphs, i.e. the study of (topological properties of) ...
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243 views

Reshetikhin-Turaev and links with a distinguished component

Hi, This question came up to me when reading the paper of Cartier on Vassiliev invariants, but it can probably be turned into a more general question. Let $T$ be the category whose objects are ...
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A knot complexity measure

Construct a knot/link by fusing two n-tangles together. (A tangle matrix shows how this might look for tangles with 6 legs. But lets use 4 legs for a start as this is far simpler.) Now, any rational ...
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674 views

Kontsevich Integral without associators?

Recall the fact that the representations of a quantum group form a braided tensor category, and this corresponds to the fact that $U_q(\mathfrak g)$ is a quasi-triangular Hopf algebra. The braiding ...
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342 views

A Category of Knot Diagrams

A brief explanation of my motivation before I ask my question. I am trying to understand Skein relations, the Jones polynomial, and their relations to Khovanov homology. To me, the natural setting to ...
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116 views

Spectral decomposition of R matrix -> Wenzl projectors?

Just curious: if you take a R matrix from knot theory and apply a spectral decomposition (see. e.g. my following post Matrix decomposition the other way) you'll get projectors: ...
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Number of the Reidemeister moves needed to transform one diagram into another one

A recent question Random Reidemeister moves to unknot contains a link to the paper http://www.ams.org/journals/jams/2001-14-02/S0894-0347-01-00358-7/S0894-0347-01-00358-7.pdf, in which J. Hass and J. ...
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Around the Montesino-Nakanishi 3-move

I have a few questions around the 3-move. I know it's NOT an unknotting move (but who needs knots with 20+ crossings anyway :-) by the recent proof of Przytycki. 1. In another paper about the third ...
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666 views

Random Reidemeister moves to unknot

Suppose one has a link diagram of the unknot, and applies random Reidemeister moves until the unknot is reached. Surely it requires an exponential number of moves, exponential in, say, the crossing ...
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506 views

Recovering the Alexander Polynomial from Ocneanu's HOMFLYPT

Let $H_q(n)$ denote the Hecke algebra associated to the symmetric group $S(n)$: this is the $\mathbb{Z}[q^{\pm 1/2}]$ algebra generated by $T_1, \ldots, T_{n-1}$ satisfying the braid relations along ...
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Untangling a graph

Assume you have a 4-valent graph (i.e., a knot universe, i.e. a collection of self-intersecting curves). Your allowed moves are the equivalents of Reidemeister 1, 2, 3, just with 4-nodes instead of ...
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398 views

Is it possible to reliably generate a particular approximation of an ideal knot via a simulated annealing approach?

Say I take a cord, tie a loose knot in three-dimensional space, and pull tightly on the ends to generate an approximation of an ideal knot. If the cord has a fixed knot topology and a random initial ...
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776 views

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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326 views

Knot polynomials: Skein>Matrix>Group?

OK, the heading was a bit tersely formulated... If you have a quantum group and an irrep, you theoretically know the R matrix (mathematicians are a notoriously idle lot, they give the general formula ...
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647 views

Examples of Non-algebraic Fibered Knots?

I am currently reading a monograph by Jose Seade, " On the topology of isolated singularities in analytic spaces". I have following questions but before asking questions I recall the definition of ...
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389 views

How does one relate the monodromy of the KZ equations with the WRT representation of the braid group?

The KZ equations on the configuration space of $n$ distinct points in $\mathbb C$ give rise to a representation of $B_n$ on $V^{\otimes n}$, where $V$ is any given representation of $SL(2)$ (we'll ...
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567 views

What are the statistics of prime knots in 3d Random walk?

This question on physics stackexchange http://physics.stackexchange.com/questions/12973/the-entropic-cost-of-tying-knots-in-polymers has a formulation which is perhaps more appropriate for this forum. ...
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Torus knots in Euclidean space — a symmetry argument

Consider a $(p,q)$ torus knot $K$ in 3-dimensional Euclidean space $\mathbb R^3$ where $p,q \geq 2$ and $\operatorname{GCD}(p,q)=1$. Let $\operatorname{Isom}(\mathbb R^3,K)$ be the isometries of ...
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What is the complex structure on the boundary torus of a hyperbolic knot complement?

Let $K$ be a hyperbolic knot in $\mathbb S^3$. Restrict the corresponding representation $\pi_1(\mathbb S^3\setminus K)\to\operatorname{PSL}(2,\mathbb C)$ to the fundamental group of the boundary ...
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Is the category of tangles that includes, X, Y, and Lambda a free Frobenius braided category?

Consider the category whose objects are non-negative integers that are represented as dots along a line, and whose morphisms are generated by $X$---positive crossing, $\bar{X}$ --- negative crossing, ...
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Is the complete functorial structure for Khovanov--Lee homology known?

I'm interested in Lee's modification of Khovanov homology, which I'll denote $\operatorname{Kh}_{\operatorname{Lee}}^\ast$. Below $L$ is a link in $\mathbb R^3$. The groups ...
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subset embedding gives trefoil knot [closed]

Let $X$ be a topological space and $E_n(X)$ the space of finite sets of cardinality $\leq n$. It is a theorem of Bott that $E_3(S^1)=S^3$. What is the idea to show that the embedding ...
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What is the three-dimensional hyperbolic volume of a four-manifold?

Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds. ...
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Knot Numerology

EDIT: Ok, I condense it to only that what is needed. Assume it's possible to use the method described here Matrix decomposition the other way to decompose a $S$ matrix from knot theory. Then each ...
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555 views

Trace identities and the Kauffman Bracket skein module

Let's consider $K_t(M)$, the Kauffman bracket skein module (see this and this papers) of a three-manifold $M$. When $t=-1$, $K_t(M)$ is easily seen to be isomorphic to the ring of functions on the ...
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Invariants of unframed, oriented links from Reshetikhin Turaev construction

Hello! I have a few questions on Reshetikhin Turaev invariants. By RT any ribbon category ${\mathcal C}$ yields an invariant of oriented, framed links labelled with objects of ${\mathcal C}$. Is ...
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Are there any very hard unlinks?

This question is closely related to a question of Gowers: Are there any very hard unknots? . I'm thinking about how to create interesting knots from small numbers of local moves on unlinks. The ...
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How many definitions are there of the Jones polynomial?

Even with the connection to quantum groups being made clearer (I believe it was not known when the Jones polynomial was first introduced), it seems to me that still we don't have the "right" ...
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A tangle matrix

The image should already say everything. But in case it is mute... :-) List tangles with 2n legs downward as column header, and with 2n legs upward as row header. (n=3 here.) In the crossing point ...
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is twisted torus knot T(7,2;4,1) prime?

what kind of twisted torus knot is prime? even more , is twisted torus knot T(7,2;4,1) prime?
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A question about Dehn filling in the unknot.

Hello, If you have a link of 2 components $U$ and $V$ in $S^3$. $U$ is the unknot, and you make $1/q$ dehn filling in $U$, you can visualize the resulting knot $V'$ from $V$, by twisting it $q$ ...
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Now that I got a mutant-discriminating invariant…

...what can I do with the darn thing? Background: I read that still no Vassiliev Invariant with mutant-discriminating power is known (correct me if this is outdated). Now, my research lead to a whole ...
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333 views

Some Vassiliev Invariant Questions

The V.I. definition goes doublepoint=overpass-underpass (or was it the other way around? If it's 50:50, I score 0 always :-). Would it lead anywhere to define doublepoint=overpass+underpass? (Even if ...
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Does every knot contain all four vertices of an isosceles trapezoid?

I ask this question with some trepidation, because it may be trivial and/or of entirely recreational interest. Erika Pannwitz proved in 1933 that every non-trivial knot contains a quadrisecant (four ...