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.
917
questions
4
votes
1
answer
225
views
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 "...
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 ...
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 ...
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 ...
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{...
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 ...
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 ...
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 ...
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 ...
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 ...
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}\...
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 ...
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'}...
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 $\partial A = -(K_0) \cup K_1$.
...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $...
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 ...
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 ...
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^{\...
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 ...
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:
...
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 ...
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 ...
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, ...
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).
...
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 ...
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 $(0,0,0)$, ...
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 ...
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 $P(2m+1,2n,2k+1)$, $m,k\ge 0 , n \ge 1$ ?
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=\...
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 (...
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 ...
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 ...
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 ...
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 ...
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 ...
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-...