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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### Is every element of $Mod(S_{g,1})$ a composition of right handed Dehn twists?

Let $S_{g,1}$ be the surface of genus $g \geq 1$ and $1$ boundary component. Let $Mod(S_{g,1})$ be the mapping class group in which we allow isotopies to rotate the action on the boundary (...

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### effectively distinguishing knots

It was proven, I think by Mijatovi? EDIT: by Waldhausen, that there is an effective algorithm for distinguishing knot complements (the effective constants were found by Coward and Lackenby). The bound,...

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### Is there a knot invariant robust to hiding one part of the diagram behind another?

Is there a well-known knot invariant that can be computed solely by inspecting an arbitrary projection of the knot into the plane (with a marking of each crossing as "over" or "under")?
The reason ...

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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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### 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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### 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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### 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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### 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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### 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 $sl(n)$ ...