All Questions
Tagged with knot-link knot-theory
35 questions
50
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 ...
- 45k
17
votes
2
answers
406
views
Random rings linked into one component?
Let $S$ be a sphere of unit radius.
Let $C_n$ be a collection of unit-radius circles/rings whose centers
are (uniformly distributed)
random points in $S$, and which are oriented (tilted) randomly (...
- 151k
12
votes
1
answer
679
views
Revisiting Gordon-Luecke theorem
$\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\GL{GL}$Here is an proof-sketch of a strengthened Gordon-Luecke theorem. This is presumably known, but is it written down somewhere? I am also ...
- 44.3k
10
votes
2
answers
598
views
Is there a "simplest" way to embed a graph in 3-space?
I consider embeddings of graphs into 3-space with edges embedded as arbitrary curves. In the simplest (non-trivial) case the graph $G$ is a cycle or union of cycles, in which case the embeddings can ...
- 13.6k
10
votes
2
answers
388
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 ...
- 151k
9
votes
1
answer
235
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 ...
- 443
8
votes
0
answers
151
views
Is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above?
For a given $N$, is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above?
The Two Summands Conjecture states that surgery ...
- 6,962
7
votes
0
answers
130
views
Generalized Brunnian links
A Brunnian link of order $n$ is nontrivial link of $n$ rings
that becomes a trivial link of $n-1$ rings if any ring is
removed. They were classified up to link-homotopy by
Milnor in 1954. This ...
- 50.8k
6
votes
1
answer
251
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 ...
- 13.6k
6
votes
1
answer
148
views
Knotted concordances of slice links
Are there any examples of a link $L$ such that:
$L$ is (strongly) slice, meaning that there exists a properly embedded collection $C$ of $n=|L|$ disjoint annuli in $S^3\times [0,1]$ such that $C\cap ...
- 193
6
votes
0
answers
211
views
$\mathbb{Z}/2\mathbb{Z}$ coefficients in gysin sequence
I am reading the article "Signature of links" by Kauffman and Taylor. Here they show that it is possible to calculate the nullity of a link $L\subset S^3$ by knowing the first betti number of the ...
- 521
5
votes
1
answer
232
views
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?
- 1,174
5
votes
1
answer
136
views
Actions of two types of Kauffman skein categories
Consider the quotient of the monoidal category of framed tangles by one of the two skein relations
together with the twist and dimension relations
Here $1_\mathbb{1}$ denotes the identity morphism ...
- 618
5
votes
0
answers
328
views
Are there examples of different knots with identical Jones polynomials and different Seifert Genus?
I had asked this question on math.stackexchange 2 days back but came up empty handed so I wanted to ask it here.
Are there known examples of $2$ non equivalent knots that have identical jones ...
- 2,689
4
votes
1
answer
262
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 ...
- 6,353
4
votes
0
answers
113
views
Coloured Jones polynomial of the mirror image of a multicomponent link
This question has been reposted from MathStackExchange
It is well understood that the usual Jones polynomial of a knot or link can be related to the Jones polynomial of the mirror image of the knot/...
- 81
3
votes
1
answer
75
views
A uniform upper bound for the linking number of periodic orbits of algebraic vector fields
Inspired by these two posts on knots orbits of polynomial vector fields on $\mathbb{R}^3$(A polynomial vector field on $\mathbb{R}^3$ which has a knot periodic orbit) and (Are total curvature and the ...
- 356
3
votes
1
answer
141
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,...
- 2,558
2
votes
1
answer
145
views
Are all Torus Links in fact Lorenz links or not?
I'm currently trying to work through the material on Lorenz knots in the literature and there seems to be conflicting information.
On p. 66, in the Birman-Williams' paper Knotted Periodic Orbits in ...
- 318
2
votes
1
answer
248
views
Wrapping a suitcase with large rotational symmetry
This is a follow-up question to Can I wrap a suitcase with hair ties.
Now we know that it is possible to wrap a suitcase with hair ties without tying them together,
but can you do it with large ...
- 45k
2
votes
2
answers
185
views
An equivalence relation on knots similar to concordance
Let $L_1$ and $L_2$ be two nonintersecting picewise-linear or smooth knots in $\mathbb R^3$. Suppose they are ambient isotopic. Does there exist an embedded surface $f: S^1\times[0,1]\to \mathbb R^3$ ...
- 2,934
2
votes
1
answer
244
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 ...
- 73
2
votes
1
answer
262
views
Numerical computation of the second Vassiliev invariant, and the permutation $(1 3 4 2)$
$\DeclareMathOperator\SLL{SLL}$For a smooth embedding $\gamma(t):\mathbb{S}^1\rightarrow\mathbb{R}^3$, the Vassilev invariant of degree 2, which I will denote as $\nu_2(\gamma)$, may be computed ...
- 149
2
votes
1
answer
164
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 ...
- 73
2
votes
1
answer
222
views
Link invariants distinguishing components
I was recently thinking about links where each component plays the same role: for every permutation of components, there is an isotopy permuting these components in the prescribed way. In the vein of ...
- 1,225
2
votes
1
answer
200
views
0-framed smoothly slice knot that can be obtained by blowing down successively a link of unknots
A knot in $S^3$ is called a smoothly slice knot if it bounds a smoothly embedded 2-disk in $D^4$. Every ribbon knot is known to be a smoothly slice knot, and there are known some nontrivial smoothly ...
- 335
2
votes
1
answer
142
views
Is there a Kauffman bracket invariant of colored links?
I want to distinguish between links where the components have different (or same) colors.
In the Alexander polynomial we can assign a different variable to each component, but what about a Kauffman ...
- 1,465
2
votes
0
answers
44
views
Link invariants on a thickened surface
Let $\Sigma$ be an oriented surface. I want to know about link invariants in $\Sigma\times [0,1]$. I already know the Ozawa polynomial introduced in this paper, but I couldn’t find any other than that....
- 21
2
votes
0
answers
80
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 ...
- 73
2
votes
0
answers
135
views
Möbius cross energy in $S^3$?
Let $\gamma_i$, $i=1,2$ be two loops in $\mathbb R^3$. The Möbius cross energy of the pair is defined by
$$
E(\gamma_1, \gamma_2)=\iint_{S^1\times S^1}\frac{|\gamma'_1(u)|\cdot|\gamma'_2(v)|}{|\...
- 2,623
2
votes
1
answer
275
views
Regular projection of a link, proof in the smooth category
Given two $C^1$ immersed curves $f, g: S^1 \to {\mathbb R}^3$ with disjoint image, I would like a simple proof, working only in the smooth category, that there exists a unit direction $y \in {\mathbb ...
- 1,123
0
votes
1
answer
122
views
Homogeneous links and crossings smoothing
Let $L$ be an oriented homogeneous link and let $D$ be an oriented diagram of $L$ wich is not necessarily a homogeneous diagram. Fix some crossing $c$ in $D$ and construct the diagram $D_0$ by ...
- 205
0
votes
1
answer
205
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 ...
- 73
0
votes
0
answers
181
views
How does the extra rope length of this link/tangle scale with the inner triangle size?
The symmetric chiral link made of three long intertwined/linked/tangled flexible ropes of radius 1 shown in the figure, whose 6 ends all lie in a plane at spatial infinity and which are pulled ...
- 9
0
votes
0
answers
61
views
Is the 3d writhe of ideal knots proportional to their smallest possible 2d writhe?
In a knot, the (two-dimensional) or 2d writhe is the sum of all positive crossings minus the sum of all negative crossings. The 2d writhe is always an integer. There is also, for each knot, a smallest ...
- 9