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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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 ...
Charles's user avatar
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11 votes
0 answers
224 views

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 ...
Tetsuya Ito's user avatar
5 votes
0 answers
88 views

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 ...
Adam's user avatar
  • 2,370
2 votes
0 answers
78 views

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 ...
Industrialactivity's user avatar
8 votes
2 answers
410 views

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 ...
yoyo's user avatar
  • 487
6 votes
1 answer
225 views

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 ...
M. Winter's user avatar
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1 vote
0 answers
56 views

"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 "...
Hauke Reddmann's user avatar
5 votes
1 answer
315 views

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 ...
MathBug's user avatar
  • 258
2 votes
0 answers
74 views

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 ...
Muqing Cao's user avatar
2 votes
1 answer
143 views

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 ...
Muqing Cao's user avatar
0 votes
0 answers
104 views

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{...
Andrea Marino's user avatar
0 votes
1 answer
203 views

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 ...
Muqing Cao's user avatar
1 vote
1 answer
241 views

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 ...
Haldot's user avatar
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9 votes
2 answers
549 views

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 ...
amd1234's user avatar
  • 333
3 votes
1 answer
208 views

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 ...
Terry Black's user avatar
9 votes
1 answer
220 views

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 ...
mmen's user avatar
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4 votes
1 answer
169 views

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}\...
Anwesh Ray's user avatar
3 votes
1 answer
192 views

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 ...
horned-sphere's user avatar
1 vote
0 answers
156 views

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'}...
Student's user avatar
  • 5,008
2 votes
1 answer
193 views

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 $\partial A = -(K_0) \cup K_1$. ...
Max Schumann's user avatar
3 votes
0 answers
96 views

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 ...
user173611's user avatar
9 votes
1 answer
259 views

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 ...
Hollis Williams's user avatar
8 votes
1 answer
617 views

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 ...
Anubhav Mukherjee's user avatar
7 votes
1 answer
419 views

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 ...
Sprotte's user avatar
  • 1,065
3 votes
1 answer
166 views

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 ...
Tristan Phillips's user avatar
2 votes
1 answer
192 views

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 ...
berto's user avatar
  • 31
7 votes
0 answers
245 views

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 ...
Allen Knutson's user avatar
2 votes
1 answer
190 views

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 ...
Muqing Cao's user avatar
5 votes
0 answers
117 views

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 $...
Misha Verbitsky's user avatar
14 votes
4 answers
2k views

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 ...
Keshav Srinivasan's user avatar
1 vote
0 answers
37 views

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 ...
Hauke Reddmann's user avatar
8 votes
0 answers
158 views

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^{\...
Adel M's user avatar
  • 113
2 votes
1 answer
263 views

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 ...
WhenYouHaveNoClue's user avatar
3 votes
1 answer
297 views

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: ...
KnotEnthusiast's user avatar
3 votes
1 answer
128 views

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 ...
Adam's user avatar
  • 2,370
2 votes
0 answers
106 views

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 ...
KnotEnthusiast's user avatar
3 votes
1 answer
157 views

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, ...
Sophia Tevosyan's user avatar
5 votes
0 answers
180 views

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). ...
Adam's user avatar
  • 2,370
3 votes
0 answers
181 views

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 ...
Minkowski's user avatar
  • 571
7 votes
1 answer
290 views

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 $(0,0,0)$, ...
Mikhail Patrakeev's user avatar
9 votes
4 answers
1k views

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 ...
Adam's user avatar
  • 2,370
1 vote
1 answer
233 views

Alexander polynomial of the pretzel knot $P(2m+1,2n,2k+1)$

Is there a (closed) formula for the Alexander polynomial of the pretzel knot $P(2m+1,2n,2k+1)$, $m,k\ge 0 , n \ge 1$ ?
Yusuf Gurtas's user avatar
2 votes
1 answer
172 views

A definition of linking number for knots in $S^3$ using chains in $D^4$

I meet this problem when reading Rolfsen's Knots&Links. After giving 8 different definitions of linking number for knots in $S^3$, he left an exercise: Given disjoint PL knots $J$ and $K$ in $S^3=\...
Math Diego's user avatar
1 vote
0 answers
81 views

The position of complex points on the kiwi-graph of the Jones polynomial

Consider the "kiwi" graph below (the name came from the resemblance to the bird, the national symbol of New Zealand, the country of V. Jones) i.e., roots of the Jones polynomial for knots (...
knotMJ's user avatar
  • 71
6 votes
1 answer
441 views

4-manifold obtained from a ribbon disk exterior by attaching a 2-handle is simply-connected if its boundary is a homology sphere

I am reading Lemma 2.1 of this paper (https://arxiv.org/pdf/2012.12587.pdf) and I can't see why $W$ is simply-connected. Here is the situation: Let $K$ be a ribbon knot in $S^3$; it bounds a ribbon ...
user302934's user avatar
3 votes
0 answers
82 views

Integral homology $S^1\times S^2$'s smoothly bounding integral homology $S^1\times B^3$'s

Suppose we are given a compact orientable 3-manifold $M$ which is an integral homology $S^1\times S^2$. Then is there a way to determine whether $M$ bounds a smooth compact orientable 4-manifold which ...
user302934's user avatar
48 votes
2 answers
2k views

Can I wrap a suitcase with hair ties

Is there a nontrivial link in a big solid torus that is trivial in the ambient Euclidean space such that each circle is unknot and has a sufficiently small length? It is motivated by a question that ...
Anton Petrunin's user avatar
3 votes
0 answers
142 views

Chekanov-Eliashberg Legendrian DGA with positive grading?

I was just looking back to some notes that I took a few years ago, when I was reading Etnyre's notes on Legendrian Contact Homology in $\mathbb R^3$ and I happened upon the following question that I ...
Nikhil Sahoo's user avatar
  • 1,175
7 votes
1 answer
435 views

Classification of knots in solid torus

What is known about the classification of knots in a solid torus $S^1 \times D^2$? Is enumerating them a reasonable problem? Do we get a similar classification as for knots in $S^3$? Ideally there ...
Calvin McPhail-Snyder's user avatar
8 votes
0 answers
375 views

The figure eight knot complement in $S^3$

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. Exercise 5.4 (on page 101) gives us a presentation of the fundamental group of $S^3 - K$ where $K$ is the figure-...
T ghosh's user avatar
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