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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 ...
guest's user avatar
  • 93
6 votes
1 answer
122 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 ...
Alessio Di Prisa's user avatar
10 votes
1 answer
299 views

Do triple-linked graphs exist?

Lets say that a finite simple graph $G$ is (intrinsically) fully triple-linked if for each embedding of $G$ into $\Bbb R^3$ we can find three disjoint cycles $C_1,C_2,C_3\subset G$ whose embeddings ...
M. Winter's user avatar
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4 votes
0 answers
73 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/...
hopftype's user avatar
0 votes
0 answers
176 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 ...
Claudio's user avatar
5 votes
0 answers
256 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 ...
Sidharth Ghoshal's user avatar
0 votes
0 answers
59 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 ...
Claudio's user avatar
2 votes
1 answer
213 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 ...
Anton Petrunin's user avatar
5 votes
1 answer
226 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?
ThorbenK's user avatar
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0 votes
0 answers
78 views

Is there any functoriality of Stallings' twists?

Suppose $L$ is an oriented link and $L'$ is a link obtained from the Stallings' twists. Are there any functoriality of the twist operation on the links, such as functoriality between (1) homology ...
jhbaik's user avatar
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6 votes
1 answer
229 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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2 votes
0 answers
78 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
154 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
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
9 votes
1 answer
226 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
  • 443
2 votes
1 answer
211 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
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
8 votes
1 answer
510 views

How to prove the product of Whitehead manifold and $\mathbb{R}$ is homeomorphic to $\mathbb{R}^4$?

I am currently reading Rolfsen's "Knots and Links". At page 82 Whitehead manifold $W$ is defined and an exercise asking to show that $W\times \mathbb{R}\cong \mathbb{R}^4$ is left. Reference ...
Math Diego's user avatar
2 votes
1 answer
180 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 ...
user302934's user avatar
2 votes
2 answers
177 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 $f: S^1\times[0,1]\to \mathbb R^3$ ...
Dmitrii Korshunov's user avatar
15 votes
4 answers
798 views

Unlinked interlocking planar polygons

Let $P$ and $Q$ be the boundary segments of two planar simple polygons. View these boundaries as rigid wires. Fix $Q$ in, say, the $xy$-plane, and imagine $P$ arranged in $\mathbb{R}^3$ so that $P$ ...
Joseph O'Rourke's user avatar
8 votes
0 answers
150 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 ...
Arshak Aivazian's user avatar
11 votes
1 answer
590 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 ...
Ryan Budney's user avatar
  • 43.5k
6 votes
1 answer
139 views

Embedding linklessly embeddable graphs without Borromean rings

A linklessly embeddable graph is a graph which can be embedded into $\Bbb R^3$ so that no two of its cycles are linked. For example, the Petersen graph is not such a graph. Now, I can think of another ...
M. Winter's user avatar
  • 12.8k
1 vote
0 answers
52 views

Mac Lane-like condition for intrinsically linked graphs?

If any embedding of your graph in 3-space has two cycles that are linked, then your graph is intrinsically linked (such as the Petersen graph). These graphs generalise non-planar graphs since for ...
ben macintosh's user avatar
3 votes
0 answers
57 views

What's the Milnor's link group for the trivial knot in a lens space?

For a link $L$ in a 3-manifold $Y$, Milnor's paper "Link Groups" https://link.springer.com/content/pdf/10.1007/BF01393902.pdf defined the link group as some quotient of $\pi_1(Y-L)$. If $L$ ...
Faniel's user avatar
  • 653
2 votes
0 answers
131 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)|}{|\...
J. GE's user avatar
  • 2,593
5 votes
1 answer
134 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 ...
Alistair Savage's user avatar
2 votes
1 answer
131 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 ...
Jake B.'s user avatar
  • 1,435
7 votes
0 answers
125 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 ...
Richard Stanley's user avatar
6 votes
0 answers
205 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 ...
Diego95's user avatar
  • 511
2 votes
1 answer
205 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 ...
Nikhil Sahoo's user avatar
  • 1,205
4 votes
0 answers
175 views

Borromean Lines of three $\mathbb{R}^1$ in $\mathbb{R}^3$ and analogous Milnor link invariants

It is know that Borromean rings can be detected by Milnor invariant $$ \bar{\mu}(\gamma_1,\gamma_2,\gamma_3)= \# (\Sigma_1 \cap \Sigma_2 \cap \Sigma_3)-\frac{1}{2}\sum_{I,J,K}\epsilon_{IJK} \sum_{\...
annie marie cœur's user avatar
5 votes
1 answer
771 views

Non-zero winding number on a space curve implies a linked curve in the zero set?

The following statement has been largely improved from my original post thanks to discussions with @DmitriPanov ad well as the comment from @Wojowu. Let $f \colon \mathbb{S}^3 \to \mathbb{R}^2$ be ...
Hao Chen's user avatar
  • 2,541
9 votes
1 answer
510 views

Does there exist a discrete gauge theory as a TQFT detecting the figure-8 knot?

My question: Does there exist a discrete gauge theory as TQFT detecting the figure-8 knot? By detecting, I mean that computing the path integral (partition function with insertions of the knot/...
wonderich's user avatar
  • 10.4k
2 votes
0 answers
55 views

Flat or linkless embeddings of graph with fixed projection

The problem of finding whether a given planar diagram of a graph, with over- and under-crossings, is a linkless embedding or not has unknown complexity (Kawarabayashi et al., 2010). My first question ...
Herng Yi's user avatar
  • 221
6 votes
1 answer
235 views

Reference request: Can iterated torus links be mutated?

I believe that most iterated torus links cannot be changed non-trivially by a Conway mutation, as follows. If you look at the JSJ decomposition of the double-branched cover, then each satellite torus ...
Dylan Thurston's user avatar
3 votes
1 answer
140 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,...
David G. Stork's user avatar
10 votes
2 answers
384 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 ...
Joseph O'Rourke's user avatar
4 votes
1 answer
253 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 ...
Christopher King's user avatar
0 votes
1 answer
115 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 ...
Mohammed Sabak's user avatar
2 votes
1 answer
142 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 ...
asldjk's user avatar
  • 318
2 votes
1 answer
251 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 ...
Lucas Seco's user avatar
  • 1,123
2 votes
0 answers
109 views

Compare two topologies: Three 2-tori inside $S^3 \times S^1 \# S^2 \times S^2$ glued from two different diffeomorphisms

We like to ask for the comparison of two topologies of three 2-tori inside the same 4-manifolds glued from two different diffeomorphisms (see the end). Given an embedded torus $T$ with trivial normal ...
wonderich's user avatar
  • 10.4k
17 votes
2 answers
399 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 (...
Joseph O'Rourke's user avatar