[HN Gopher] A fast, strong, topologically meaningful and fun kno...
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A fast, strong, topologically meaningful and fun knot invariant
Author : bikenaga
Score : 27 points
Date : 2025-09-25 15:07 UTC (3 days ago)
(HTM) web link (arxiv.org)
(TXT) w3m dump (arxiv.org)
| m_dupont wrote:
| To their credit, the paper is unexpectedly fun.
|
| I'm not understanding the "separation power" thing, what does
| that imply?
| gjm11 wrote:
| They mean that their invariant does a good job of
| distinguishing different knots from one another.
|
| The way they quantify this is: they pick a biggish set of knots
| that are known all to be distinct from one another. They then
| compute their invariant for each of those knots. A knot
| invariant successfully distinguishes them all from one another
| precisely when it takes different values for all of the knots.
| So they count the number of different values their invariant
| takes, and subtract it from the number of knots. They call this
| the "separation deficit": the smaller the better.
|
| They compare their invariant with some already-known ones,
| taking "all knots that can be drawn in the plane with <= 15
| crossing points" as their set of knots. There are about 300,000
| of these.
|
| One of the best-known knot invariants is the so-called
| Alexander polynomial. That's in row 3 of Table 5.1, and its
| "separation deficit" for those knots is on the order of 200k.
| That is, these 300k knots have between them only about 100k
| different Alexander polynomials; if you pick a random smallish
| knot and compute its Alexander polynomial then handwavily you
| should expect that there are two other different smallish knots
| with the _same_ Alexander polynomial.
|
| Another knot polynomial, which does a better job of
| distinguishing different knots, is the so-called HOMFLY
| polynomial. (Why the weird name? It comes from the initials of
| the six authors of the paper announcing its discovery.) That's
| row 7, showing a deficit of about 75k. That suggests, even more
| handwavily, that if you pick a random smallish knot and compute
| its HOMFLY polynomial, there's about a 1/3 chance that there's
| another smallish knot with the same HOMFLY polynomial. Still
| not great.
|
| A rather different sort of invariant is the _hyperbolic volume
| of the complement of the knot_. That is: if you take _all of
| space minus the knot_ then there 's a certain nice way to
| define distances and volumes and things in the left-over space;
| the whole of space-minus-the-knot turns out then to have a
| finite volume, and perhaps surprisingly deforming the knot
| doesn't change that volume. So that's another knot invariant,
| and it turns out to be better at distinguishing knots from one
| another than the polynomials mentioned above, on the order as
| 2x better than the HOMFLY polynomial.
|
| This paper's invariant (which is a pair of polynomials) does
| about 6x better than what you get by looking at the Alexander
| polynomial, the HOMFLY polynomial, the hyperbolic volume, and a
| few other invariants I didn't mention above, all together. Its
| "separation deficit" on this set of ~300k knots is about 7000.
| If you pick a random smallish knot, there's only about a 2%
| chance that some other knot has the same value of this paper's
| invariant.
|
| (Reminder that all this business about probabilities is super-
| handwavy. Actually, that probability might be anywhere from
| about 2% to about 4% depending on exactly how the values of the
| invariant are distributed.)
|
| Now, all of this is purely empirical and looks only at smallish
| knots. So far as I know they haven't proved any theorems like
| "our invariants do a better job than the hyperbolic volume for
| knots with <= N crossings, for all N". I think such theorems
| are very hard to come by.
|
| They don't, to be clear, claim that their invariant is the
| _best_ at distinguishing different knots from one another. For
| instance, they mention another set of knot polynomials that
| does a better job but is (so they say) much more troublesome to
| compute for a given knot.
| QuadmasterXLII wrote:
| What would people recommend to get up to speed on the alexander
| polynomial? The wikipedia page was terse as expected.
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