implement Conway polynomial

[fchapoton]

as deduced from Alexander polynomial

CC: @tscrim @miguelmarco @slel @soehms

Component: algebraic topology

Keywords: knot

Author: Frédéric Chapoton

Branch/Commit: bd2b791

Reviewer: Sebastian Oehms, Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/27522

[fchapoton]

comment:1

still needs work

---

New commits:

7130792

implement the Conway polynomial of knots and links

[fchapoton]

Branch: u/chapoton/27522

[fchapoton]

Commit: 7130792

[fchapoton]

comment:2

There remains an issue about normalisation, plus more doctests needed.

[embray]

comment:3

As the Sage-8.8 release milestone is pending, we should delete the sage-8.8 milestone for tickets that are not actively being worked on or that still require significant work to move forward. If you feel that this ticket should be included in the next Sage release at the soonest please set its milestone to the next release milestone (sage-8.9).

[fchapoton]

Changed keywords from none to knot

[soehms]

comment:5

I don't understand the line

M = M // 2

in your code. If I uncomment it I'm getting the correct result.

BTW: Wouldn't it by useful, to introduce a second option to obtain the Conway polynomial via the Homfly polynomial for cross check purpose?

sage: B = BraidGroup(3)
sage: L41 = Link(B([1, -2, 1, -2]))
sage: h41 = L41.homfly_polynomial(normalization='az'); h41
L^2 - M^2 - 1 + L^-2
sage: c41 = L41.conway_polynomial(); c41
1 - t^2
sage: t, = c41.variables()
sage:  hc41 = h41(1,t); hc41
1 - t^2
sage: hc41 == c41
True

Since now we have Puiseux series available the Homfly polynomial can also easily be used for cross checks with Alexander and Jones polynomials:

sage: a41 = L41.alexander_polynomial(); a41
-t^-1 + 3 - t
sage: j41 = L41.jones_polynomial(); j41
t^2 - t - 1/t + 1/t^2 + 1

sage: P.<t> = PuiseuxSeriesRing(ZZ)
sage: ha41 = h41(1,t^(-1/2) -t^(1/2)); ha41
-t^-1 + 3 - t
sage: ha41.laurent_series().laurent_polynomial() == a41
True

sage: hj41 = h41(~t,t^(-1/2) -t^(1/2)); hj41
t^-2 - t^-1 + 1 - t + t^2
sage: hjp41 = hj41.laurent_series().laurent_polynomial()
sage: L = hjp41.parent()
sage: hjp41 == L(j41)
True

As soon as we have Puiseux polynomials available, an option to use it would be a nice enhancement here, too.

[soehms]

Changed branch from u/chapoton/27522 to u/soehms/27522

[soehms]

Changed commit from 7130792 to 9f0bbf3

[soehms]

Reviewer: Sebastian Oehms

[soehms]

comment:7

Frederic, I still think that the line of code I posted in my previous comment (from June 2020) should be removed. After removing it the cross check against all of KnotInfo's Conway polynomials passes.

sage: B = BraidGroup(3)
....: L = Link(B([1, -2, 1, -2]))
....: L.conway_polynomial()
-t^2 + 1
sage: R = _.parent()
sage: kicps = {k:R(k[k.items.conway_polynomial].replace('z', 't')) for k in KnotInfo}
sage: len(kicps)
7166
sage: %time all(k.link().conway_polynomial() == kicps[k] for k in KnotInfo)
CPU times: user 6min 25s, sys: 270 ms, total: 6min 26s
Wall time: 6min 26s
True

I push a corresponding fix and in addition add some doctests. If you agree with it you may give it a positive review.

---

New commits:

e0c61a3	Merge branch 'u/chapoton/27522' of trac.sagemath.org:sage into conway_polynomial_27522
9f0bbf3	27522: fix an some more doctests

[tscrim]

comment:8

One small comment:

-: wikipedia:`Alexander_polynomial`
+:wikipedia:`Alexander_polynomial`

[sagetrac-git]

Changed commit from 9f0bbf3 to d5f83fd

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

d5f83fd

27522: typo

[soehms]

comment:10

Replying to @tscrim:

One small comment:

-: wikipedia:`Alexander_polynomial`
+:wikipedia:`Alexander_polynomial`

Fixed!

[fchapoton]

comment:11

looks good to me. Please set to positive if you agree

[fchapoton]

comment:12

oh, by the way please fix the syntax in the second meth role here:

Sebastian Oehms (October 2020, add get_knotinfo() and meth:isisotopic

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

9cfb578	Merge branch 'u/chapoton/27522' of trac.sagemath.org:sage into conway_27522
05e98be	Merge branch 'u/soehms/27522' of trac.sagemath.org:sage into conway_27522
d40f741	Merge branch 'u/soehms/27522' of trac.sagemath.org:sage into conway_27522
bd2b791	27522: fix syntax error in meth role

[sagetrac-git]

Changed commit from d5f83fd to bd2b791

[soehms]

comment:14

Replying to @fchapoton:

looks good to me. Please set to positive if you agree

Thanks!

oh, by the way please fix the syntax in the second meth role here:

Done.

[soehms]

Changed reviewer from Sebastian Oehms to Sebastian Oehms, Travis Scrimshaw

[vbraun]

Changed branch from u/soehms/27522 to bd2b791