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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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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Topological description of inverting a knot

The first figure shows an offset overhand knot. To tie it, you simply bring the two cords together and make an overhand knot in them. It's more secure than it looks, and several climbers have been ...
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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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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 $f(k,1) \in \{1,2,\ldots,n\} \times \{1\}$....
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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 $2$-...
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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 ...
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Alexander polynomial in branched covers

Suppose I am given a 3-manifold as a double branched cover over a link. Let a null-homologus knot in this space be given as a lift of an arc with endpoints on the link (it is automatically null-...
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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 ...
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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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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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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 $\...
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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 ...
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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 ...
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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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What fraction of knots are ribbon knots?

Pardon the naive question. Perhaps there are several different variants of my question, but essentially I am seeking to know whether ribbon knots are vanishingly rare among all knots, or a positive ...
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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 $I^3$, ...
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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 $0\leq k\leq n/2$. \begin{align} P_{0,0}&=1\\ \text{for $n\geq 1$}...
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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 ...
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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 ...
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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 example:https://www.youtube.com/...
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Stable, tame A-infinity isomorphism?

Linear duality provides a correspondence between differentials on free tensor algebras and A-infinity algebras. Under this correspondence, what is the A-infinity counterpart of "stable, tame ...
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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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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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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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Distribution of Random Knots from Braids

Let $R_{2n,l}$ be a random braid word of length $l$, where each letter is chosen uniformly from the braid generators of $B_{2n}$, $\{\sigma_1,\ldots,\sigma_{2n-1},\sigma_1^{-1},\ldots,\sigma_{2n-1}^{-...
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Essential surfaces in knot complements

Given any knot $K \subset \mathbb{S}^3$, one can find a closed oriented embedded surface $S$ such that $K \subset S \subset \mathbb{S}^3$. Moreover, pick such an $S$ that has minimal genus. One can ...
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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 $\sigma_{1},...,\sigma_{n}$....
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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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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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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 ...
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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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Expressing knot polynomials as casimirs

I always wondered why one writhe unit (read: colored with the irrep of a quantum Lie algebra and evaluated as Reshetikhin-Turaev invariant thereof) is essentially the quadratic casimir of that irrep. (...
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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 Sutcliffe....
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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, ...
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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. ...
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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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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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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 ...
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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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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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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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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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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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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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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 \mathbb{C}...
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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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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 http://en.wikipedia.org/wiki/...
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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 ...
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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 ...