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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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 ...

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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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### 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 ...

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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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### 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 ...

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### 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 ...

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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 ...

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### 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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### 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 ...

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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 $sl(n)$ ...

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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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### 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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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 ...

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**1**answer

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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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 ...

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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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### “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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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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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 ...

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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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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 ...

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### 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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### 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. ...

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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 ...

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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 ...

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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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**1**answer

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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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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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### 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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**1**answer

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 ...

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**1**answer

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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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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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### 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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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 ...

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**1**answer

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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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**1**answer

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 ...

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**1**answer

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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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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 ...

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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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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 ...

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**1**answer

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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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**1**answer

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 ...

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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 ...