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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Are there knots that can be distinguished by the Alexander-Conway polynomial, but not the Alexander polynomial?
On page 9 of Kauffman's Formal Knot theory, Kauffman claims
The Alexander-Conway Polynomial is a true refinement of the Alexander Polynomial. Because it is defined absolutely (rather than up to ...
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1answer
160 views
Modules over Hopf Algebras and $E_2$-algebras
Justin Young has a paper on the brace bar-cobar duality between hopf algebras and $E_2$-algebras: https://arxiv.org/pdf/1309.2820.pdf
I was wondering if anybody knows of a nice relationship between ...
6
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1answer
150 views
Do an unlinked trefoil and figure-eight cobound an annulus in $B^4$?
Let $K_1$ the trefoil (left or right hopefully does not matter?) and let $K_2$ be the figure-eight knot in $S^3 = \partial B^4$. Are there any smooth properly embedded annulus $A$ in $B^4$ with $\...
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0answers
52 views
Is there a measure of the failure of the Alexander polynomial to distinguish knots?
Has there been any research into something like the ratio of distinct Alexander-indistinguishable knots to total knots (up to some measure of complexity)? This was a random question asked of me by a ...
13
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1answer
80 views
Which knots are singularities of a hyperbolic cone-manifold structures on $S^3$?
Which knots $K\subseteq S^3$ are such that there is a hyperbolic cone-manifold structure on $S^3$ that has (exactly) $K$ as the singular locus?
What if in Question 1 we restrict the cone angles to be $...
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35 views
Common invariants of virtual knots?
When I google invariants of virtual knots (links), there's a bunch of polynomial (and other) invariants, but it's very hard to distinguish which of these are considered as somewhat classical.
In the ...
2
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1answer
85 views
Basis for Annular Skein Algebra
Background/Notation:
Given the Iwahori-Hecke algebra $H_{n}$ (over some ring commutative ring $R$ with identity) with generators $\{T_{1},\ldots T_{n-1}\}$, we know it has basis
$\{T_{w}\}_{w\in S_{...
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IH-moves on trivalent graphs, and a complex that might be known to low-dimensional topologists
Here is a combinatorial problem which is hard to Google but seems like it might have a solution well known to people who study finite type invariants etc.
Let $G_{g,b}$ denote the set of finite ...
6
votes
1answer
208 views
HOMFLYPT vs. Jones vs. Alexander polynomial?
I'm searching for examples (perhaps the simplest one?) to show that the HOMFLYPT polynomial is stronger than the Jones and Alexander polynomial, respectively.
Any ideas what is the 1st knot in the ...
5
votes
0answers
81 views
Relation between different versions of Bar-Natan homology
In Bar-Natan's paper: Khovanov’s homology for tangles and cobordisms, he defined a deformation of Khovanov homology. Namely, for any $m\geq 0$, Bar-Natan's homology $BN^{m}(K)$ is obtained by ...
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0answers
95 views
Brylinski Beta Function Calculation
I've recently read the paper written by Brylinski on the Beta function of a knot, where he gave the example of a trivial "circular" knot. Having a physics background, and not being formally introduced ...
3
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1answer
187 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 $sl(n)$ ...
4
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1answer
95 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 ...
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0answers
56 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.
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1answer
83 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
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1answer
77 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 ...
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0answers
145 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
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2answers
205 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-...
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vote
1answer
102 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 ...
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3answers
112 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 $\...
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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
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0answers
154 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 $X,Y$...
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1answer
192 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
233 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 ...
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0answers
20 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
146 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
139 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
130 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,...
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0answers
137 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]...
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votes
1answer
411 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
126 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 $J(x,t)=(\phi_t(x),t)$. ...
48
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 ...
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votes
2answers
238 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 $\phi(0)$...
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votes
1answer
200 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
185 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
119 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
799 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 ...
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0answers
52 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
304 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
94 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.
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0answers
125 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 ...
7
votes
1answer
292 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
121 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 ...
6
votes
1answer
222 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 ...
8
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0answers
111 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
189 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
131 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
253 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
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1answer
379 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 ...
