# Tag Info

### What part is left unsolved in the Unknotting problem? (after results of Bar-Natan, Khovanov, Kronheimer and Mrowka)

To strengthen Sam Nead's answer, note that it is trivial to compute the Jones polynomial from the Khovanov homology. It is known that computing (or even approximating) the Jones polynomial is #P-hard: ...
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### Why should I care about the Jones polynomial?

Your question presupposes that people were excited about the Jones polynomial because it would help them to classify/distinguish knots. In fact, I suspect the interest came from the fact that this ...
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### What part is left unsolved in the Unknotting problem? (after results of Bar-Natan, Khovanov, Kronheimer and Mrowka)

EDIT: Marc Lackenby has just announced a quasi-polynomial time algorithm. That is, given an $n$—crossing diagram, the algorithm takes $n^{O(\log(n))}$ time to either find a spanning disk (proving the ...
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### How to add essentially new knots to the universe?

Yes, forcing can add fundamentally new knots, not equivalent to any ground model knot. Indeed, whenever you extend the set-theoretic universe to add new reals, then you must also have added ...

### Why should I care about the Jones polynomial?

As a historical note (others may have had a different perspective - I was a graduate student when the Jones polynomial made its appearance), when it came out there was some mild excitement because the ...
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### Can I wrap a suitcase with hair ties

This configuration should work: Edit (to provide credit/context): Michael Freedman's solution (see Ian Agol's post) is the original one. Ian directed me to this problem and gave me the hint that ...
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### What part is left unsolved in the Unknotting problem? (after results of Bar-Natan, Khovanov, Kronheimer and Mrowka)

This isn't directly what you ask, but it's also worth noting that unknot detection is in $\text{NP} \cap \text{co-NP}$, that is, there are polynomial-checkable certificates that will show that either ...
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### Unknot recognition - how tangled does it get?

Joel Hass and Jeff Lagarias proved that one can transform any unknot diagram with $n$ crossings into the standard unknot diagram using not more than $2^{cn}$ Reidemster moves. They were able to obtain ...
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### $S^3$ as cyclic branched cover of itself

The statement that for arbitrary K in $S^3$, if for some $n \ge 2$, the n-fold cyclic branched cover is $S^3$ (or in some versions, a homotopy 3-sphere) then K is the unknot, was known as the Smith ...
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Yes, this is done in Penney, D.E., Generalized Brunnian links, Duke Math. J. 36, 31-32 (1969). ZBL0176.22201. Call a link $(n,k)$-Brunnian if it has $n$ components, and every sublink with $m$ ...
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### Why should I care about the Jones polynomial?

There have been some topological applications of the Jones polynomial and its various generalizations. I believe that these applications increased the interest in these invariants by topologists. ...
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Here is a figure of a (4,2)-Brunnian link (in the terminology of Mark Grant's answer): And here is an image of a (5,3)-Brunnian link: These are taken from G.C. Shephard's 2006 article "Interlinked ...
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