# Questions tagged [knot-theory]

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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### Canonically representing the monodromy of a hyperbolic manifold fibered over $S^1$

Let $Y$ be a hyperbolic manifold that fibers over $S^1$, with fibration $\pi:Y \to S^1$ with fiber $\Sigma$. Thurston states that the monodromy $\phi:\Sigma \to
\Sigma$ of this projection is then ...

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### References for algorithms for computing knot invariants

I'm wondering if there's compiled literature on well-known algorithms and their bounds for computing various knot invariants (I'm writing a Master's thesis on the subject).
I can find individual ...

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### Finite application of one of Reidemeister moves on a knot diagram

It is known that given a knot diagram we can transform it into a trivial unknot diagram by a series of Reidemeister moves. The key word is "series".
Can we transform any knot diagram using a ...

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### Abelian covering of link complement

I'm considering finite index abelian (regular) covering of link complement:
$$ X \rightarrow S^3\setminus L$$
where $L$ is a minimally twisted chain link.
I'm interested in covering space. Can we ...

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### Explicit parameterizations of complicated unlinks?

I have a somewhat empirical question which I hope is still welcome here. I would like to know how to write down explicit parameterizations of "complicated unlinks", say with 2 or 10 ...

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### Covering of a knot complement

Let $B=S^3\setminus K$ for some (tame) knot $K$. Suppose we have a covering $E\to B$ with a finite fiber.
Question: is $E$ homeomorphic to a knot/link complement?
On this question I found only the ...

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### Amenable link groups

The unknot and the Hopf link are (as far as I know) the only links whose complements have abelian fundamental groups. Are there more examples whose complement have amenable fundamental group?

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### Corollary in Rasmussen's paper about $s$-grading of Lee's canonical generators

In Jacob Rasmussen's paper Khovanov homology and the slice genus, he states as Corollary 3.6 that $s(\mathfrak s_o)=s(\mathfrak s_{\bar o})=s_{min}(K)$, where $s$ is the $s$-grading and $\mathfrak s_o,...

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### Bounds for the crossing number in terms of the braid index?

Is there a lower bound on the crossing number of a knot (resp., link) with braid index $b$?
For knots, I believe the smallest crossing number for braid index 2 is 3, the smallest crossing number for ...

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### The second coefficient of the Conway polynomial from Knot Floer homology

Let $\nabla_K(z)$ be the Conway polynomial and $\Delta_K(t)$ be the Alexander polynomial normalized by $\Delta_K(t)=\Delta_K(t^{-1})$ and $\Delta_K(1)=1$,
These invariants are equivalent and they are ...

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### Irreducible factors of the A-polynomial

The A-polynomial $A_K$ of a knot $K$ describes the irreducible "non-abelian" components of the $SL(2)$-character variety of $S^3-K.$
Does anyone know a knot K for which $A_K$ has more ...

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### Cubic lattice representation of a solid torus knot using the surface as a boundary

For physics simulation reasons, I would like to respresent a solid torus knot as a collection of integer points sat on a cubic lattice.
If I were to do this using a sphere, I would do this by saying ...

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### Knot Diffie–Hellman

Here's an idea for a knot-based Diffie–Hellman exchange:
Public: random (oriented) knot $P$.
Private: random (oriented) knots $A$ and $B$.
Exchange: Alice sends (randomized or canonical ...

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### In knot theory, what is this link property and how to detect it: "linkings between components separate nicely"

The following could be made more general (see below), but let's focus on a link $L$ that consists of three components (closed curves) $\gamma_1,\gamma_2,\gamma_3\subset\Bbb R^3$.
Call $L$ a necklace ...

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### "Higher" knot mutants

Mutation Wiki My related question 1 My related question 2
Top: How wiki describes mutation. Doesn't generalize well.
Bottom: How I think of it.
Now replace "four" in the Wiki text by "...

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### Are two slice surfaces with minimal genus isotopic?

For a link $L\subset S^3$ and two Seifert surfaces (edit: a better name would be slice surfaces as the comments below 1 2 point out) with minimal genus $S_1,S_2\subset B^4$, I have the following ...

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### Composition of 3-braids to obtain braids with trivial closure

Given a 3-braid $b=\sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1}\sigma_1$ (which has non-trivial closure), can we find a 3-braid $c$, which has trivial closure (closure results in any trivial knot or ...

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### Non-straightenable multiple space-time trajectories and 'entangled' braid

Considering we have the space-time trajectory of multiple particles (or any objects) in the X-Y-Time coordination system. Given a projection direction parallel to the X-Y plane, we can obtain the ...

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### Are there infinite number of 3-braids with trivial closure?

Not counting equivalent braids, are there finite or infinite numbers of 3-braids whose closures are trivial knot or links? If the answer is infinite, are there some patterns in those infinite numbers ...

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### Name for homotopy totalization of Goodwillie tower (in embedding calculus)

Let $M,N$ be a manifold and consider the presheaf of spaces $\textrm{Emb}(-, N)$ on the open sets of $M$. Classical embedding calculus produces a goodwillie tower
$$ \ldots \rightarrow T_{k+1} \textrm{...

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### Space-time trajectory that cannot be straightened and its braid form

Considering we have the space-time trajectory of multiple particles (or any objects) in the X-Y-Time coordination system. Given a projection direction, we can obtain the braid form of the space-time ...

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### Ways to prove that $n$-component Brunnian link is nontrivial

The attached image shows a way to construct an $n$-component Brunnian link for any $n\geq 3$. That is, this link is not trivial, but deleting any of its components makes the new link trivial. The ...

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### Bing sling isotopy to unknot

Rolfsen asked the question as to whether any knot is topologically isotopic to the unknot. Where a topological isotopy is a continuous path in $\operatorname{Emb}(S^1,\mathbb{R}^3)$.
From now on I ...

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### Picturing twisting of strands explicitly after blow downs

In order to simplify Kirby calculus proofs, one can use a box notation which indicates a number of full twists up to sign. In the scenario of two strands with twist boxes, it is straightforward to ...

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### Links and non-orientable surfaces

Let $\Sigma \subset \mathbb{R}^3$ be a compact embedded surface with boundary $\partial \Sigma$ and $i:\Sigma\setminus \partial\Sigma \to \mathbb{R}^3 \setminus \partial\Sigma$ the inclusion.
Is the ...

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### Alexander polynomials for a certain family of closed braids

Let $n\geq 3$ be a positive integer and $\kappa=(k_1, \dots, k_n)\in \mathbb{Z}^n$. Denote by $B_n$ the braid group on $n$ strings. Consider the braid on $n+1$ strings $\sigma_\kappa:=\sigma_1^{k_1}\...

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### Algebraic variations of the full knot Floer complex

In Hom's paper (arXiv link), p.20, Section 3.3 ends with
"There are other algebraic modifications one may consider, such as setting $U^n =
0$ or $UV = 0$",
referring to the knot Floer ...

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### Khovanov $A_\infty$ algebra

Let $L$ be a link in $\mathbb{R}^3$, with $D, D'$ be diagrams in
$\mathbb{R}^2$ representing $L$. Khovanov constructed two graded
chain complexes $$C_{D} = (Ch_{D}, d_{D}) \quad C_{D'}=(Ch_{D'},
d_{D'}...

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### Knot concordance, hyperbolicity and amphichirality

Let $K_0$ and $K_1$ be two knots in $S^3$. We say $K_0$ and $K_1$ are concordant if there exists a smoothly embedded annulus $A \subset S^3 \times [0,1] $ such that $\partial A = -(K_0) \cup K_1$.
...

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### Kauffman bracket for Abelian anyons

The Kauffman bracket, defined here, assigns a polynomial in $A$ to any knot. (For concreteness consider the Kauffman bracket normalized so that the unknot is assigned $-(A^{-2}+A^2)$.) For certain ...

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### Physics application of Wilson surface observables

There is some work which generalises the usual Wilson loop in QFT to higher dimensions and constructs non-abelian Wilson surface functionals in the context of non-abelian gerbes.
It seems to me that ...

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### On trivial mapping class group of 3-manifolds

What are some examples of knots $K\subset S^3$ such that the mapping class group of $S^3_{1/n}(K)$ is trivial? I guess for hyperbolic knots with no symmetry in the complements are good candidate as ...

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### Small knots becoming isotopic after connect sum

I am interested in the following situation: I have two codimension-2 knots $K_1$ and $K_2$ in $S^n$ and they are not isotopic. Furthermore, $K_1$ is not isotopic to the mirror image of $K_2$ and ...

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### Knot group of a field extension

Notation:
$L/K$, finite extension of global fields
$K^\times$, unit group of $K$
$L^\times$, units group of $L$
$\mathbb{A}_L^\times$, ideles of $L$
$N_{L/K}$, the norm map
The knot group of an ...

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### Which hyperbolic fibered knots have monodromy with a single singularity?

The figure eight-knot has pseudo-Anosov monodromy with no singularity. I have read that the (-2,3,7)-pretzel knot has pseudo-Anosov monodromy with a single 18-prong singularity on the boundary of the ...

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### Generating cycles inside Tits' graph of words for a positive braid

Let $Br_n$ be the braid group and consider words in its generators (not in the inverses). Two such words define the same "positive" braid if one can be obtained from the other by commuting ...

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### Determine if a closed braid is a link/unlink

I am relatively new to the world of braids/knots so really sorry if this question is simple. However, I am not able to find if there is any theorem/procedure that determines if a closed braid, given ...

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### Isotopy classes of $CP^1$ in 4-manifolds

Let $S_1$, $S_2$ be homologous embedded 2-spheres
in a compact smooth 4-manifold. Under which additional
conditions are they smoothly isotopic? I am interested
in the state of the art picture when $...

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### Is there an algorithm for the genus of a knot?

A Seifert surface of a knot is a surface whose boundary is the knot. The genus of a knot is the minimal genus among all the Seifert surfaces of the knot. My question is, is any algorithm known to ...

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### Tetravalent graph invariant: Vassiliev in disguise?

Let's start with a virtual link, just that it has no over- and undercrossings, but simple nodes. Random example (A). For the virtual crossings, the usual laws hold (B). Also as usual, loose loops drop ...

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### Is the Lawrence–Krammer representation faithful, reduced modulo p?

It is well-known that the braid group $B_n$ is linear for every $n$ by the Lawrence–Krammer (or LKB) representation. It embeds $B_n$ faithfully into $\mathrm{GL}\left(\frac{n(n-1)}{2},\mathbb{Z}[q^{\...

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### Classification of congruent integer matrices

I am interested in the following question:
Let $A,B\in\text{Mat}(2n\times2n;\mathbb{Z})$ be two integer matrices with the property that $\text{det}(A-A^T)=1=\text{det}(B-B^T)$. Are there known ...

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### A faulty proof that a Whitehead Double of a knot is smoothly slice

We denote the untwisted Whitehead double of a knot $K$ to be $Wh(K)$. As an example, here is the oriented Whitehead double of the figure eight knot:
Let us look in the neighborhood of the clasp:
...

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### Characterizing algebraic tangle by their double branched covers

Montesinos proved that the double branched cover $\Sigma(T)$ of an algebraic tangle $T$ in a $3$-ball is a graph manifold. I wonder if the converse true: Is $T$ algebraic if $\Sigma(T)$ is a graph ...

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### General formula for a topologically slice odd pretzel knot

An odd 3-strand pretzel knot $K=P(p,q,r)$ has $\Delta_K(t)=1$ if $pq+pr+qr=1$. This fact, along with a theorem of Fintushel and Stern (every odd 3-pretzel knot with trivial Alexander polynomial is not ...

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### The same PD code seems to yield two different knot diagrams of the Hopf link

The PD code [(2, 3, 1, 4), (4, 1, 3, 2)] seems to map to a non-unique knot diagram. I can describe the following two Hopf links with different orientations with this same PD code. As I understand it, ...

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### Lens space to lens space surgeries

Let $M_r(K)$ denote the slope $r$ surgery on a knot $K\subset M.$
Gordon conjectured and Kronheimer-Mrowka-Ozsvath-Szabo proved that if
$S^3_r(U)=S^3_r(K)$ for some $r$ then $K=U$ (the trivial knot).
...

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### Reshuffling power series (aka Melvin–Morton expansion in knot theory)

I am struggling to understand a statement which follows from some change of variables in a power series. I think that the context does not really matter here, so I will put it at the end of the ...

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### Is it possible to separate two linked (geometric) circles in $\Bbb R^3$ by a set homeomorphic to the 2-sphere (with arbitrarily “bad” homeomorphism)?

$A$ and $B$ are two linked (geometric) circles in $\Bbb{R}^3$. (Let, for definiteness, both have radius = 1, the first lies in the $z=0$ plane and its center is the origin of coordinates $(0,0,0)$, ...

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### Which knot complements are double branched covers?

Denote the double branched cover of a $2$-tangle $T\subset B^3$ by $\Sigma(T)$. Since $\partial \Sigma(T)$ is a torus, I wonder if anyone studied the question: which knot complements in $S^3$ are of ...