13 votes
Accepted

Solving the unknotting problem by pulling both ends of the string

Here is a paper that, I think, underlines some of the difficulties in recognising the unknot using the physical process of "pulling tight". Nontrivial embeddings of polygonal intervals and ...
Sam Nead's user avatar
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6 votes

Solving the unknotting problem by pulling both ends of the string

As noted in the comments, it is difficult to make this intuitive idea mathematically precise. In practice, even slip knots get stuck on themselves when pulled tight. One may look for monotonic ways of ...
Ian Agol's user avatar
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6 votes
Accepted

Slice knots in 3-manifolds

Suppose you know that the universal cover of $Y$ embeds in $S^3$, i.e. is $S^3-A$ for some $A$. For example, this happens when $Y$ is a connected sum of lens spaces. (I'm thinking this is always true ...
Danny Ruberman's user avatar
6 votes
Accepted

Rational 4-tangles vs rational knots

The closure of a non-rational four-tangle can yield an unknot. Here is one family of examples. Start with a nontrivial two-tangle (that is, an arc embedded in a three-ball, which is not boundary-...
Sam Nead's user avatar
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6 votes
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Heegaard splitting of figure-8 knot complement

Here is a sequence of figures showing how to go from a knot diagram (for the figure-eight knot) to a Heegaard splitting of the knot exterior. Your request for “the mapping class group element” is ...
Sam Nead's user avatar
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5 votes
Accepted

Knotted concordances of slice links

I think this is likely an unknown question. Namely, the negation of 3) would follow from 1) and 2) if strongly slice links are strongly ribbon (which seems to be open) ribbon disks bounding the ...
Ian Agol's user avatar
  • 66.8k
2 votes

Solving the unknotting problem by pulling both ends of the string

I think your description of a knot as a string with endpoints pulled apart is similar to embedding knots in $\mathbb{RP}^3$, with the free endpoints corresponding to a single point on $\mathbb{R}^3$'s ...
Michael's user avatar
  • 2,175
1 vote

Is there a nontrivial ribbon knot concordance from a knot to itself?

In the topological category, a locally flat concordance from knot $K$ to knot $J$ is homotopy-ribbon when the fundamental group of $S^3-K$ injects into the fundamental group of the concordance ...
Hall's user avatar
  • 11
1 vote

Space of the trivial long knot in the thickened surface

Let us show that $\mathcal E=Emb_0(I,F\times I)\sim\Omega_0(F,x_0)$. We start with R. Budney's remark. Proposition. Let $F$ be a connected compact 2-manifold and $P(F)$ the pseudoisotopy group, i.e ...
nim's user avatar
  • 357

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