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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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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152 views

What do tangles teach us about braids?

A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times ...
10
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547 views

Categorical Schur's Lemma

In attempt to prove (and compute) a formula for the dimensions of the HOMFLY homology of the (p,q)-torus knot one could try to follow original proof by Jones of a formula for HOMFLY polynomial of ...
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166 views

The Markov trace via Bott-Samelson fibers?

Let $H_n$ be the Hecke algebra of GL(n), i.e., the algebra over $\mathbb{Q}(q)$ with generators $T_1, \ldots, T_{n-1}$ which satisfy the braid relations and also $T^2 = (q-1) T + q$. Recall the ...
9
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281 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 ...
8
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139 views

Does the shortest path between two braids pass through string links?

One of the fundamental facts underlying the application of braid theory to knot theory is that braids inject into string links. This means that braids $B_1$ and $B_2$, considered inside a cube $I^3$, ...
8
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458 views

Categorification of finite type invariants

Recall that finite type invariants are those numerical knots invariants vanishing on knots with enough singularities. The value of an invariant on a singular knots is computed using the Vassiliev ...
8
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307 views

What is (explicitly) known about the SL(n,C) character varieties of 3-manifolds ?

The $SL(2,{\bf C})$ character variety of a 3-manifold with 1-cusp $M$ (like a knot complement in the 3-sphere) essentially coincide with the variety defined by the A-polynomial. Those polynomials are ...
8
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277 views

Which presentations of (non)planar algebras give rise to knots?

Reidermeister's theorem states that the set of knots, modulo ambient isotopy, is isomorphic to the planar algebra generated by crossings, modulo Reidemeister moves. This planar algebra presentation is ...
8
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224 views

Krull rings and determinantal invariants

During another attempt to come to grips with Hillman's excellent book Algebraic Invariants of Links, I am having difficulty figuring out why Krull rings are the setting for Chapter 3- the natural ...
7
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217 views

What is the historical connection between Zeeman's twist spinning and Fox's Examples?

Both Ralph Fox and (at that time, yet to be knighted) Sir Christopher Zeeman attended the 1961 Georgia topology conference. Fox's paper from that conference was his seminal work, "A Quick Trip through ...
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452 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 $0\leq k\leq n/2$. ...
6
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251 views

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, ...
5
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116 views

Software for BMW algebra calculations?

Does software exist for computations in the BMW algebra? For example, I'd like to be able to express elements in a basis of "totally descending tangles" as in a paper of Morton–Wassermann. At ...
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104 views

Crossing bound implies Reidemeister move bound?

In 1998 Galatolo established an upper bound on the number of Reidemeister moves needed to convert a diagram $D$ of the unknot into a trivial loop diagram. The upper bound is a function of $n$, the ...
5
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211 views

Torsion in chord diagrams for 2-string links?

The abelian group of $k$-chord diagrams on a skeleton of two directed line segments (modulo the STU relation), $\mathcal A_k(\uparrow\uparrow)$, is known to have $2$-torsion when $k=5$. In fact, I ...
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82 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 ...
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100 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, ...
4
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234 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 ...
3
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117 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 ...
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123 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 ...
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264 views

On finding A-polynomials

I have two questions to obtain the explicit forms of A-polynomials. Takata used the mathematica pacage qMultisum.m to obtain the recursion relation of the colored Jones polynomials for twist knots. ...
3
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166 views

Constructing Markov traces simply

Short version: I wondering how to simply check if a proposed Markov trace, $\phi$ had the correct property using techniques similar to those from the Akutsu-Wadati 1987 paper `Exactly solvable models ...
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302 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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196 views

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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236 views

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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339 views

Positivity of braid monodromy of curve singularities

I recall the notion of braid monodromy. Let $C \subset \mathbb{C}^2$ be an algebraic curve, and choose a projection $\pi: \mathbb{C}^2 \to \mathbb{C}$ such that the restriction $\pi: C \to ...
3
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286 views

Computing Quantum Dimensions

Hi, in "Jaeger’s Higman-Sims state model and the B2 spider" by Greg Kuperberg (arxiv:math9601221v1, 1996) there are some quantum dimensions listed in the "Discussion" part. Evidently quantum groups ...
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352 views

ribbon links - counterexamples

An n-component link is said to be ribbon if it bounds a ribbon surface consisting of n discs. (a ribbon surface is an immersed surface with only ribbon singularities, see ...
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35 views

Tait conjectures for alternating w-links

The Tait Conjectures are useful in knot tabulation. For alternating knots and links, two of them state: Any reduced diagram of an alternating link has the fewest possible crossings. Any two reduced ...
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91 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?
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110 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 ...
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128 views

Simple terminology question about the Dubrovnik (Kauffman) polynomial

In my S matrix classification attempts I encounter a lot of Dubrovnik polynomials of the form D(z-1/z,z^n) and D(-z+1/z,z^n). [Second variable is for writhe, n is an integer; for the first I don't ...
2
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219 views

Which quandles arise as knot quandles

Is it known which quandles arise as the knot quandle of some knot? I seem - vaguely - to remember hearing that all (finite?) quandles arise this way but I've failed to find a reference to this fact.
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673 views

ambient isotopy and isotopy on knot

this is elementary question about classical knot equivalence. I know that just isotopy which need not to be ambient is not proper to define knot equivalence because bachelor's unknotting. but this ...
2
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278 views

Branched covering vs Covering space of 0-surgery manifold ??

Let $K$ be a knot in $S^3$ and $M^3$ be a 3-manifold obtained by 0-surgery from $S^3$ along $K$. By using Mayer-Vietoris sequence, we can see that $H_i(M^3)=H_i(S^1\times S^2)$. Therefore, we have a ...
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89 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 ...
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92 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 ...
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89 views

Can all knot polynomials derived from Skein relations be categorified?

Can all knot polynomials derived from Skein relations be categorified?
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154 views

Things you can do with the self-writhe

I hope "self-writhe" is the established word. (0 for link-crossing, otherwise identical to writhe +1 or -1) I bet the following is known: Take some crossing of a link with self-writhe $w_a$. Flip it ...
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137 views

Is there a two-variable E8 polynomial? (Conjectural or proven)

On MO I learnt about the two-variable E7 polynomial (status: conjectural). What about a two-variable E8 polynomial? I have reasons to believe such a thing exists too, but I do magic, not math, so my ...
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153 views

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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192 views

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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315 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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112 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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93 views

Alexander Invariant of Torus knot

I am very interested in knot theory, especially in knot groups and knot polynomials. As is well known, it is easy to calculate the Alexander polynomial from the fundamental group $\pi_{1}(K)$ of a ...
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83 views

Jones polynomial of 2-knots

Question: is it possible to define the Jones polynomial for knotted surfaces (or $S^2$ for simplicity) in $R^4$? Jones polynomial has several definitions (see How many definitions are there of the ...
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146 views

link group of the trivial $n$ component link

Currently I am reading Milnor's paper "Link Groups". In the paper he defines the link group, for every $n$ component link $L$, as a certain quotient of the fundamental group $\pi_1 (S^3 \setminus L)$. ...
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152 views

Finding a ribbon graph for a mapping class group action

Turaev defines TQFT $(T, \tau)$ in his book "Quantum invariants of knots and 3-manifolds". He uses it to define an action of a mapping class group of a d-surface $\Sigma$. This action $\epsilon$ is ...
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144 views

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 ...