41 votes

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: ...
Dylan Thurston's user avatar
40 votes
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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 ...
Jonny Evans's user avatar
  • 6,925
38 votes
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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 ...
Sam Nead's user avatar
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38 votes
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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 ...
Joel David Hamkins's user avatar
26 votes

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 ...
Igor Rivin's user avatar
  • 95.5k
26 votes
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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 ...
Larsen Linov's user avatar
24 votes

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 ...
Dylan Thurston's user avatar
23 votes
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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 ...
JoshuaZ's user avatar
  • 6,060
22 votes
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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 ...
Danny Ruberman's user avatar
21 votes
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Link such that deleting any two components leaves an unlink

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 $m$ ...
Mark Grant's user avatar
  • 34.9k
21 votes

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. ...
Ian Agol's user avatar
  • 66.5k
18 votes

Link such that deleting any two components leaves an unlink

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 ...
j.c.'s user avatar
  • 13.5k
18 votes
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On trivial mapping class group of 3-manifolds

Dave Gabai proved that the mapping class group of a closed hyperbolic 3-manifold is isomorphic to its isometry group. For a hyperbolic knot $K$ without any symmetries, for large enough $n$, $S^3_{1/n}(...
Ian Agol's user avatar
  • 66.5k
17 votes

Applications of arithmetic topology to number theory

Le and Murakami (HERE and HERE) discovered several previously unknown relations between multiple zeta values through the study of quantum invariants of knots. Further relations were later discovered ...
Daniel Moskovich's user avatar
17 votes

Elementary proof that knot complements are path-connected

A proof may be given along the lines of the proof of the Jordan Curve theorem by Doyle (see this answer). This uses the fundamental group and a variation on Van Kampen, but not homology. So this ...
Ian Agol's user avatar
  • 66.5k
17 votes
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Can Khovanov homology have arbitrarily large torsion?

This paper from earlier this year (Jan 18, to be precise) proves the existence of $\mathbb{Z}/n\mathbb{Z}$-torsion for $n\le 8$ and $\mathbb{Z}/2^s\mathbb{Z}$-torsion for $s\le23$. It also states at ...
Marco Golla's user avatar
  • 10.3k
17 votes
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Simple question on Kirby move

Yes, there is a simple way. Below is a sequence of pictures illustrating the procedure (created using Kirby calculator). $5_2$: Blowup at the clasp: Isotopy: Blowdown the purple unknot:
Marco Golla's user avatar
  • 10.3k
17 votes

What is the state of research on finding all prime knots with 17 crossings?

Ben Burton has found that there are 352,152,252 prime non-trivial knots with up to 19 crossings. See here for the tables. 2022-06-11 update: The details of this enumeration have now been published in ...
Josh Howie's user avatar
  • 1,617
14 votes

How to motivate the skein relations?

One of the earliest appearances of the ingredients for a skein relation can be found in Romilly Allen's 1904 book on Celtic Knotting. There he explains that designers of Celtic knot patterns first ...
Louis H. Kauffman's user avatar
14 votes
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Classification of knots by geometrization theorem

You have all the tools to compute the geometric decomposition of knot and link exteriors in the software Regina. I'm one of the authors, although my hands haven't been over that part of the code very ...
Ryan Budney's user avatar
  • 42.7k
14 votes
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Which knots are singularities of a hyperbolic cone-manifold structures on $S^3$?

The results proved in S. Kojima, "Deformations of hyperbolic 3-cone-manifolds", J. Differential Geom. 49 (1998), no. 3, 469-516 provide complete answers to questions 1 and 3. The main theorem of ...
Roberto Frigerio's user avatar
14 votes

Several questions about Gauss's mathematical conception of braids

• Connection with electromagnetism: (see Gauss' linking number revisited for the historical context) Consider a wire $c$ carrying a current $I$, winding around a closed loop $c'$, as in the ...
Carlo Beenakker's user avatar
13 votes

Elementary proof that knot complements are path-connected

Not an answer to the question, but a hopefully related observation to complete the proof and explain why the result is not obvious, which is may be of interest for your class. I would mention that a ...
Pietro Majer's user avatar
13 votes

SL(2, C)-representation of a knot

$(P)SL(2, \mathbb{C})$ is the isometry group of $\mathbb{H}^3,$ so $SL(2, \mathbb{C})$ representations are the natural generalization of hyperbolic structures on knot complements.There is a vast ...
Igor Rivin's user avatar
  • 95.5k
13 votes

$0$-surgeries on trefoil and figure-eight

They can also be distinguished geometrically. Both knots are genus one fibered knots, so both $M$ and $N$ are torus bundles over the circle. The complement of the figure eight is hyperbolic, so the ...
Josh Howie's user avatar
  • 1,617
13 votes
Accepted

Why is the thing dual to a "meridian" called a "longitude"?

There is a fundamental asymmetry between latitude and longitude on a sphere, whereas on a torus, there is a symmetry between the two generators. This symmetry could motivate the use of nearly ...
Timothy Chow's user avatar
  • 77.8k
13 votes

Is there an algorithm for the genus of a knot?

There is an algorithm using normal surface theory, originally developed by Haken and Schubert to compute the genus of any knot. These articles are in German, but for a reference in English, one could ...
Josh Howie's user avatar
  • 1,617
13 votes
Accepted

Is there an algorithm for the genus of a knot?

Jaco and Oertel's paper An algorithm to decide if a three-manifold is a Haken manifold [1984], plus a bit of work, gives a doubly exponential time algorithm to compute the Seifert genus. (In practice ...
Sam Nead's user avatar
  • 25.3k
12 votes

Does the union of all finite groups yield a complete knot invariant for prime knots?

Though it is not completely obvious, it turns out that if $G_1$ and $G_2$ are finitely generated groups that surject onto the same set of finite groups, then the profinite completions of $G_1$ and $...
Andy Putman's user avatar
  • 43.1k
12 votes
Accepted

Rational slice knot that is not slice

Yes. The figure-eight knot is an example: it bounds a smooth slice disk in a rational homology ball. This has been proven in a bunch of different ways, going back to the 1980s. Here are a couple of ...
Adam Levine's user avatar

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