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.
3
votes
1answer
176 views
Framing dependence of HOMFLY polynomial
I want to understand the framing dependence of the Khovanov-Rozansky homology, and as its first step, I am trying to understand the framing dependence of the HOMFLY polynomial (i.e. quantum $sl(n)$ ...
3
votes
1answer
76 views
Quandle homomorphism does not always induces group homomorphism on inner automorphism groups of quandles
Let $X$ and $Y$ be two quandles and $f: X \rightarrow Y$ be a quandle homomorphism. Then we can define a map $\bar f: Inn(X) \rightarrow Inn(Y)$ as $\bar f(S_a)=S_{f(a)}$, where $a \in X$. Then $\bar ...
2
votes
0answers
48 views
Blow-up and Blow-down kirby local moves for non-orientable $3$-manifold
Can anyone explain or give a reference about the Blow-up and Blow-down Kirby local moves for non-orientable $3$-manifolds?
Thanks, advance.
3
votes
0answers
54 views
Multivariate Alexander polynomial vs single variable (Conway) Alexander polynomial
I consider the multivariate Alexander polynomial $\Delta(t_1,\ldots,t_n)$ for a $n$-component link (defined using e.g. the Fox derivative).
If we wish to construct a 1-variable polynomial $A(t)$, we ...
2
votes
1answer
73 views
How are characterstic polynomials (resp. Alexander polynomials) distributed amongst adjacency matrices (resp. grid diagrams)?
Fix $n$, and consider the characteristic polynomials for all $C=2^{\frac{n(n-1)}{2}}$ adjacency matrices representing undirected, unweighted graphs on $n$ vertices.
Are the characteristic ...
9
votes
0answers
129 views
Define the 3d Chern-Simons TQFT on a discrete simplicial complex
Question: What is the challenge and the current status to define the 3d Chern-Simons(-Witten) (CSW) theory on a simplicial complex or on a discrete lattice? (Or is there a no-go or an obstruction ...
10
votes
2answers
183 views
Is the Lisca-Matic bound (aka slice-Bennequin bound) strictly stronger than the Bennequin bound?
The Bennequin bound [1] says that, for a transverse knot (or later link) $K$ in $S^3$,
$$\mathrm{sl}(K) \le - \chi(\Sigma)$$
for any Seifert surface $\Sigma$ for $K$, where $\mathrm{sl}$ is the self-...
1
vote
1answer
101 views
“Flat links”, a reference request
A hyperbolic link is one whose complement admits a hyperbolic metric.
Hyperbolic links, and especially hyperbolic knots, are quite popular these days. However, I am currently interested ...
2
votes
2answers
89 views
Multivariable vs single variable Alexander polynomial for links?
If we take a $n$-component link $L$, we have the multivariable Alexander polynomial $\Delta(L)(t_1,\ldots,t_n)$. Is there a standard single-variable Alexander polynomial? If yes, is it just euqal to $\...
31
votes
3answers
2k views
Why should I care about the Jones polynomial?
The invention of the Jones polynomial led to hundreds of papers and a Fields medal. However, as far as I can tell it had few consequences in topology. After all, after Thurston’s work we already ...
2
votes
0answers
144 views
What is an ambient isotopy categorically?
Let $\mathcal T$ be a category of "nice" topological spaces (CW?) and continuous maps between them. We can construct the homotopy category $\mathrm{Ho}\mathcal{T}$ which gives for any two objects $X,Y$...
6
votes
1answer
184 views
What is the complexity of determining if a knot group is $\mathbb{Z}$?
It is known from the work of Waldhausen that the isomorphism problem for knot groups is decidable. What is then:
The complexity of determining if a knot group is $\mathbb{Z}$? .i.e. same as the ...
9
votes
3answers
222 views
Knots of fixed genus with arbitrarily large volume
Consider all knots with fixed genus $g\ge 2$ (I am considering the classical 3-genus). Do there exist infinite families of genus $g$ knots with arbitrarily large volume?
The answer seems like it ...
0
votes
0answers
17 views
Maximum possible crossing number of a restricted tangle family
Let $T$ be a tangle (in the sense of knot theory) on $s$ strands and no loops. In general the crossing number of $T$ is unbounded, as we can twist two strands around each other any number of times. ...
5
votes
1answer
141 views
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 ...
4
votes
1answer
130 views
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 ...
3
votes
1answer
128 views
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,...
6
votes
0answers
132 views
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]...
22
votes
1answer
404 views
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 ...
3
votes
1answer
108 views
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 $J(x,t)=(\phi_t(x),t)$. ...
49
votes
2answers
2k views
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 ...
4
votes
2answers
235 views
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 $\phi(0)$...
4
votes
1answer
193 views
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 ...
5
votes
1answer
171 views
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 ...
4
votes
1answer
111 views
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) ...
15
votes
2answers
788 views
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 ...
1
vote
0answers
48 views
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 ...
10
votes
2answers
298 views
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 ...
1
vote
1answer
93 views
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.
9
votes
0answers
123 views
Interactions between pseudoline arrangements and braid groups?
It is common to represent
pseudoline arrangements
as wiring diagrams:
Fig. from: "Hamiltonicity and colorings of arrangement ...
6
votes
0answers
213 views
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 ...
3
votes
1answer
106 views
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 ...
5
votes
1answer
215 views
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 ...
7
votes
0answers
106 views
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 ...
7
votes
1answer
149 views
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=...
2
votes
2answers
179 views
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 ...
3
votes
1answer
129 views
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 ...
2
votes
1answer
252 views
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 ...
11
votes
1answer
377 views
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 ...
7
votes
1answer
310 views
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 ...
0
votes
1answer
101 views
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 $S^3$" (...
4
votes
1answer
151 views
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 ...
2
votes
1answer
126 views
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 ...
2
votes
1answer
153 views
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) := ...
13
votes
1answer
388 views
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 ...
1
vote
1answer
90 views
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 ...
3
votes
0answers
128 views
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 ...
4
votes
0answers
291 views
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 ...
5
votes
1answer
121 views
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 ...
4
votes
3answers
251 views
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. ...
