Questions tagged [knot-theory]
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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Show a sequence of sums involving Catalan Numbers converges
Let $C_n$ be the $n$-th Catalan Number and let $\mathcal{O}_{s,j} = {{2s-j-1}\choose{j}} C_{s-j}^2$. Then we want to consider $\mathcal{E}_s = \sum_{j=0}^{s-1} (-1)^j\mathcal{O}_{s,j}$. We want to ...
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How disconnected can a Seifert surface be?
Seifert surfaces
The standard definition of a Seifert surface for a link in $S^3$ is an oriented, compact surface embedded in $S^3$, bounding the link. Often, it is assumed to be connected, but given ...
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Links defined by link-severance tableau
Consider a finite $n$-element classical (real) link and the resulting link structure obtained by cutting each of the component elements (knots). Let us represent the resulting structures in a tableau,...
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Theory of Irrational Tangles?
According to one possible definition, an $n$-tangle $T$ is a subset $T \subseteq \Bbb{R}^2\times [0,1] =: X$ that is homeomorphic to a disjoint union $[0,1] \times n := [0,1] \amalg \ldots \amalg [0,1]...
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Tying knots via gravity-assisted spaceship trajectories
Q.
Can every knot be realized as the trajectory of a spaceship
weaving among a finite number of fixed planets, subject to gravity alone?
To make this more ...
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Isotopy extension theorem: how non-unique is ambient isotopy
Let $M$ and $N$ be smooth manifolds. Consider an isotopy of $M$ inside $N$. This means that we have a level preserving embedding $J\colon M\times [0,1] \to N \times [0,1]$. Put $J(x,t)=(\phi_t(x),t)$. ...
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How to add essentially new knots to the universe?
A knot is an embedding of a circle $S^{1}$ in $3$-dimensional Euclidean space, $\mathbb{R}^3$. Knots are considered equivalent under ambient isotopy. There are two different types of knots, tame and ...
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Why are two diffeomorphims of $R^n$ are always homotopic (in the same category)?
Where can one find the proof of the following fact:
If there are two orientation-preserving diffeomorphisms $\phi_0$ and $\phi_1$ of $R^n$, then there exists a homotopy $\phi(t)$, such that $\phi(0)$...
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Conformal boundary and cusp of figure-8 complement
As we know the figure-8 ($4_1$) complement can be obtained by quotienting $\mathbb{H}^3$ with an arithmetic Kleinian group, which has index 12 inside $PSL(2,\mathcal{O}_3)$. The resulting complete ...
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Knot Factorization Homology inputs
Following the paper by Ayala, Francis, and Tanaka: https://arxiv.org/pdf/1409.0848.pdf
If we are talking about knots we are talking about framed 3-manifolds with a framed 1-dimensional sub-manifold ...
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Presentations of the monoidal categories of virtual tangles and of welded tangles by generators and relations
Reidemeister theorem implies, without too much fuss, that the monoidal categories of tangles, and of oriented tangles, can be presented by generators and relations. This is done for example in
a) ...
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Link such that deleting any two components leaves an unlink
Brunnian links are well known, where deleting any component allows you to isotope the rest to an unlink. It's common to construct them by taking an $n-1$ component unlink and defining the $n$th ...
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Resolving a mismatch in indexing conventions of knot/link Floer homologies
I have trouble matching the indexing conventions for Ozsvath-Szabo's knot Floer homology with link Floer homology.
Say we have a knot $K$ in a 3-sphere. Then we can consider the filtered chain ...
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Tangled random triangles: One giant component?
Suppose you have $n$ triangles whose corners are random points on a sphere $S$
in $\mathbb{R}^3$.
Viewing the triangles as built from rigid bars as edges,
two triangles are linked if they cannot be ...
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Can we compute the Tristram–Levine signatures of a knot in $S^3$ using Jacobian with Fox partial derivatives?
My question is in the tittle:
Can we compute the Tristram–Levine signatures of a knot in $S^3$ using Jacobian with Fox partial derivatives?
If the answer is yes, is there a reference for this.
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Interactions between pseudoline arrangements and braid groups?
It is common to represent
pseudoline arrangements
as wiring diagrams:
Fig. from: "Hamiltonicity and colorings of arrangement ...
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Does annular Khovanov homology detect the unknot (in annulus)?
Recently Kronheimer and Mrowka showed that Khovanov homology detects the unknot. It's still not known if the Jones polynomial detects the unknot.
Does annular Khovanov homology detect the unknot in ...
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Version of Khovanov homology that does not produce torsion?
Khovanov homology usualy has torsion. But there are also different versions of Khovanov homology. Is there a Khovanov homology theory that naturaly does not produce torsion?
A followup question: can I ...
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Generating prime knots (in order)
In this really cool paper https://arxiv.org/abs/1612.03368, A. Malyutin shows that the probability that a random prime knot of up to $N$ crossings (as $N$ goes to infinity) is not generically ...
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Slicing satellite knots
Call a knot L a "braid-pattern satellite" of a knot K if it is a satellite of K and the pattern on which it is based is a closed braid in the solid torus. Is there a knot K so that no braid-pattern ...
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Factor traces of the Temperley-Lieb algebra
Given $\delta\in\mathbb C$, let $A(\delta)$ denote the complex unital $*$-algebra generated by an identity $1$ and selfadjoint elements $e_k$, $k\in\mathbb N$, satisfying $e_k^2=\delta e_k$, $e_ke_l=...
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Minimum required crossings in a link diagram for a $k$-component Brunnian link
What is the minimum number, $s$, of crossings in a link diagram for $k$ (component) links fully knotted together such that cutting any single link frees all individual component links--becomes an ...
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Infinitely many Brunnian links with bounded crossings
A set of Brunnian link is a nontrivial link such that if one component is removed, it becomes trivial. The best known example is the Borromean rings:
Here's a six component example:
There is likely ...
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How many non-homeomorphic collections of $N$ circles in $\mathbb{R}^3$ are there?
Let's have a finite collection of $N$ circles $\mathbb{S}^1$ in $\mathbb{R}^3$. (These circles could not intersect.) Every circle could be "hooked on" other circle and it could be "hooked" for ...
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How hard is it to guess Kuperberg's certificate of knottedness?
Kuperberg's Knottedness is in $\mathsf{NP}$, modulo GRH provides a certificate that a knot $K$ given by a knot diagram on $c$ crossings is not trivial. The certificate is a prime $p$, along with a ...
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Can we cut and rotate a particular region of a hyperbolic 3-manifold to get another (non-homeomorphic) hyperbolic 3-manifold?
I'm trying to learn more about hyperbolic 3-manifolds, in particular the geometric implications of doing hyperbolic Dehn surgery to suitable knot complements.
Following this paper by Christian ...
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What does it mean exactly for a pair of $S^0$'s to be unlinked on a knot $K$?
I am trying to learn about the effects of knot mutation on the hyperbolic manifolds obtained via hyperbolic Dehn surgery, and I'm currently reading Ruberman's paper "Mutations and Volumes in $S^3$" (...
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Can a knot be cables of two different knots?
I wonder if there is an example of a knot $K$ in the 3-sphere which can be realized as cables of two distinct (up to isotopy) knots $K_1 \neq K_2$.
It is known that if a knot $K$ is the $(p,q)$-cable ...
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Classifying links with essential annuli in the complement as torus links
I've seen it claimed in several places, though never with a detailed proof, that every non-split link is either a hyperbolic, satellite, or torus link (see for example pg. 95 of Cromwell's "Knots and ...
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Fary-Milnor Theorem : Help following a proof on page 9
I am studying Fary-Milnor Theorem on total curvature of knots and I am stuck in a proof. He is proving on page 9:
The Total curvature of a tame knot cannot equal the curvature of its type
k(C) := ...
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Does Lackenby's polynomial bound on knot moves imply polynomial mixing in "Quantum Money From Knots?"
In the 2010 paper Quantum Money from Knots Farhi, Gosset, Hassidim, Lutomirski, and Shor give a doubly stochastic Markov chain acting on grid diagrams. Transitions in the Markov chain are permutations ...
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Can a planar tangle have an infinite number of input disks?
Can a planar tangle have an infinite number of input disks?
Some publications talk about cases with a finite number of input disks, while others do not say if it is finite or infinite.
So, is it ...
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Lutz twist and open book decompositions
Let $M^3$ be a closed oriented 3-manifold, endowed with an open book decomposition. Consider a section of the open book, that is a knot $K \subset M$ disjoint from the binding and meeting every page ...
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Can knot non-equivalence be a proof-of-work for a cryptocurrency?
Regarding a question about proofs-of-work and following up on this answer and the comments therein, I believe we can, at least in theory, come close to having the hashing resources used in ...
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Pull-back of knots in branched covers and the Alexander polynomial
Given a knot $K \subset S^3$ one can form its double branched cover $\Sigma_2(K)$ and consider the pull-back knot $\widetilde{K} \subset \Sigma_2(K)$ of $K$ to $\Sigma_2(K)$ (the locus fixed by the ...
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Given a link $L\subset S^3$ how to construct a link $L'$ whose complement have hyperbolic structure?
Thurston claimed that almost all closed 3-manifolds are hyperbolic. To support this, he said that every closed 3-manifold is obtained by Dehn surgery along some link whose complement is hyperbolic. ...
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What happens if we generalize the fundamental group to make knotted loops distinct?
The definition of an element of the fundamental group of a space $X$
based at point $p$
is
$$f:[0,1]\rightarrow X,\quad f(0)=p=f(1),$$
defined up to homotopy.
This homotopy allows self-intersection,
...
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Is there a zero knowledge protocol for knottedness, similar to the GMW protocol for graph non-isomorphism?
In the very easy-to-read [1], Kuperberg shows that, conditioned on the Generalized Riemann Hypothesis, knottedness is in $\mathsf{NP}$. As I understand the proof, given a knot-diagram of a knot $K$, ...
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Homogeneous links and crossings smoothing
Let $L$ be an oriented homogeneous link and let $D$ be an oriented diagram of $L$ wich is not necessarily a homogeneous diagram. Fix some crossing $c$ in $D$ and construct the diagram $D_0$ by ...
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Knots with high unknotting number relative to their genus
Can the unknotting number of a knot with fixed three-genus be arbitrarily high?
Some background and motivation: the unknotting number $u(K)$ of a knot $K$ (i.e. the minimum number of necessary ...
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Twisting equivalent links and the isotopy type of the resulting links
Take any link $L_1$ with an unknotted component
$K$, cut along the disk bounded by K (which usually intersects some of the
other components of $L_1$ transversally ), twist n times, and reglue. Let us ...
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Can we "Curve" a manifold, as much as possible?
Assume that $M$ is a $k$ dimensional manifold which is embedded in $\mathbb{R}^n$. We define the map $\phi_{M,n}: M \to G(k,n)$ with $\phi_{M,n} (x)= T_x M$, the tangent space to $M$ at point $x\in M$....
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Extensions of Fried's Theorem: Surface bundles over Circles and Flows on 3 Manifolds
The Thurston Polytope provides a way to organize information about the embedded surfaces living in a 3-manifold.
His amazing theorem, often called the "Fibered Faces" Theorem, says that if you have ...
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Efficient algorithm for determining if a knot is trivial by querying its crossings
Consider the projection of a knot $K$ having $n$ intersections, such as this:
Assume we have prior information that the source knot is equally likely to be any one of the distinguishable knots having ...
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Can real algebraic knots be recovered from their projections?
Consider a knot represented by a real algebraic curve $C$ in $\mathbb{R}^3$. For example, this answer on SE gives a representation of the trefoil knot. Given a projection of $C$ onto the plane (sans ...
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Does there exist a limit of (infinite) iterated universal abelian covers of a knot group?
Let $K$ be a knot, $G=\pi_1(S^3 - K)$ and $G^{(n)}$ is the $n$-th term
of the derived series. Then $G/G^{(n)}$ is poly-torsion-free abelian since it is known that each $G^{(i)}/G^{(i+1)}$ is torsion-...
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What is the relationship between the Khovanov-Rozansky homology of a digraph and that of a link?
Motivation: I'm reading this preprint, which takes a digraph $G = (V, E)$ and then builds a projective algebraic set $P(G)$ by assigning a variable to each edge and then defining certain polynomial "...
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Twisted Whitehead double of trefoil knot
Let $K$ be a knot whose Alexander polynomial is not trivial and $G = \pi_1(S^3-K)$. By Tim Cochran's noncommutative theory, $G$ is not solvable. But is $G$ a hypoabelian group? In particular, I'm ...
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Are all Torus Links in fact Lorenz links or not?
I'm currently trying to work through the material on Lorenz knots in the literature and there seems to be conflicting information.
On p. 66, in the Birman-Williams' paper Knotted Periodic Orbits in ...
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Simple question on Kirby move
From hyperbolic volume computation, I found that the following two 3-manifolds are (possibly orientation-reversal) homeomorphic:
surgery on figure-eight knot $4_1$, with slope $-5$, and
surgery on $...