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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 $P$ ...
117 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 ...
353 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 ...
95 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 ...
45 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 ...
47 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 $L$ ...
126 views

265 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 ...
436 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/...
47 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 ...
194 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 ...
135 views

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,...
341 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 ...
190 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 ...
73 views

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 ...