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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?
• 815
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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 ...
196 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 ...
• 11.3k
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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 ...
117 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 ...
189 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 ...
195 views

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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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 ...
1k 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 ...
• 41.2k
290 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 ...
134 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 ...
• 475
166 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 $f: S^1\times[0,1]\to \mathbb R^3$ ...
• 1,324
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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
138 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 ...
• 4,790
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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
116 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 ...
• 11.3k
1 vote
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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 ...
48 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 $L$ ...
• 505
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• 2,077
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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 ...
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### 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/...
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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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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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### 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 ...
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217 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 ...
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91 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 ...
136 views