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 ...
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 ...
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 ...
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-...
6
votes
Accepted
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 ...
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 ...
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 ...
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 ...
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 ...
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