Questions tagged [knot-link]
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37
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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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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 ...
- 1
6
votes
1
answer
196
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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 ...
- 11.3k
2
votes
0
answers
57
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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 ...
- 73
2
votes
1
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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 ...
- 73
0
votes
1
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189
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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 ...
- 73
9
votes
1
answer
195
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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 ...
- 413
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votes
1
answer
151
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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 ...
- 73
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votes
2
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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 ...
- 41.2k
6
votes
1
answer
290
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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 ...
- 85
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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 ...
- 475
2
votes
2
answers
166
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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 $f: S^1\times[0,1]\to \mathbb R^3$ ...
- 1,324
15
votes
4
answers
786
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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 $P$ ...
- 147k
8
votes
0
answers
138
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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 ...
- 4,790
11
votes
1
answer
476
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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 ...
- 41.6k
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1
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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 ...
- 11.3k
1
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50
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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 ...
3
votes
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answers
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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 $L$ ...
- 505
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130
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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)|}{|\...
- 2,573
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votes
1
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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 ...
- 598
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1
answer
108
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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 ...
- 1,385
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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 ...
- 46.3k
6
votes
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answers
178
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$\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 ...
- 501
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1
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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 ...
- 1,087
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149
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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_{\...
- 2,077
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votes
1
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641
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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 ...
- 2,489
9
votes
1
answer
470
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/...
- 10.1k
2
votes
0
answers
51
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 ...
- 221
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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 ...
- 9,953
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votes
1
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138
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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,...
- 2,518
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votes
2
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368
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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 ...
- 147k
4
votes
1
answer
217
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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 ...
- 4,355
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1
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91
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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
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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 ...
- 308
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votes
1
answer
216
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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 ...
- 941
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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 ...
- 10.1k
17
votes
2
answers
382
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 (...
- 147k