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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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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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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 ...
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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.
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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 ...
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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 ...
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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 ...
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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-...
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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 ...
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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 $\...
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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 ...
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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 $X,Y$...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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,...
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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]...
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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 ...
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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 $J(x,t)=(\phi_t(x),t)$. ...
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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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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 $\phi(0)$...
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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 ...
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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 ...
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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) ...
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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 ...
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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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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=...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $S^3$" (...
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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 ...
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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 ...
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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) := ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...