Knot polynomial

A knot polynomial isparticular knot invariant. It can also be viewed ashash function. The coefficients areimportant part;polynomialnot meantbe evaluated, but merelywayindexingsetnumbers.

"Polynomial"used inmuch more general sense thanusual. As functionsx, theseactually Laurent polynomialsx^1/nvarious n.

Tablecontents

1 Justification 2 Alexander polynomial 2.1 Example 2.2 Example 2 3 Alexander-Conway polynomial 4 Jones polynomial 5 HOMFLY(PT) polynomial 6 BLM/Ho polynomial 7 Kauffman unary polynomial 8 Kauffman binary polynomial 9 Unworked examples 10 (Composing notes)

Justification

Why bother? For one thing,polynomialmuch easiercommunicate thanknot, or evendrawing ofknot.

For another, it's far easiercompare two polynomialsequivalence than two knots. Ifknot-to-polynomial mapping can be calculated from elements ofknotis sufficiently discriminating, two complicated knots can be checkedidentity algorithmically.

The latter condition ishardersatisfy.

Of course polynomialsnotonly things available; another hash onknot isleast numbercrossings needed indiagramit. But that does not discriminate knots at all well. Another hash isFukuhara/O'Hara energy, which discriminate fairly well—an energy E correspondsat most 0.264×1.658^E knots—buthardcompute.[1] actuallylooks like E increases rather rapidly, wrtcrossings, so "rather well" may be optimistic Therealsoropelength[1].

It's also possible that elementary polynomial operations could turn outhave analoguesknot manipulations. Indeed, this isidea behind skein relations.

Alexander polynomial

James W. Alexander inventedfirst useful knot polynomial1923,published1928. Technically, an Alexander polynomial isgenerator ofprincipal Alexander ideal related tohomology ofinfinitely cyclic cover ofknot complement—where allemphasised phrases have particular mathematical meanings. Fortunately there isshortcut that computespolynomial fromcrossingsan oriented knot.

Procedure, somewhat informally:

1) Numberknot's crossings, 1…N. Prepare an N×N matrix M. (Q: does any ol' diagram do, or doeshavehave minimal crossings?)

2) Walk alongknot. As you pass over crossing n,crossing p onleftcrossing q onright, add tomatrix:

3) FillrestMzeros.

4) Drop from M any one rowany one column.

5) TakedeterminantM (thisan Alexander polynomial ofknot).

6) Normalise by dropping allzero roots and, ifhighest-degree coefficientnegative, negating.

The result‘the’ Alexander polynomial ofknot.

Example

Ontrefoil knot:

knot	crossings
numbereddirected trefoil	n	p	q
	1	2	3
	2	3	1
	3	1	2

resulting inmatrix

Takeminor M₂₃

trefoil: left-handed trefoil right-handed trefoil

Example 2

Onstevedore knot:

knot	crossings
image:Knot-stevedore-numdir-128.png	n	p	q
	1	3	6
	4	6	5
	5	3	2
	6	4	1
	3	1	2
	2	4	5

to makematrix

1-x	0	x	0	0	-1
0	1-x	0	x	-1	0
x	-1	1-x	0	0	0
0	0	0	1-x	-1	x
0	-1	x	0	1-x	0
-1	0	0	x	0	1-x

resulting

figure-eight: figure-8

Suppose there isknot andplane which touchesknot at exactly two points (this may need stricting-up). The portion ofknot which lies on one side ofplane, closed withsegment joiningtwo points,another knot. The original knotsaidbesum oftwo lesser knots so formed. A knot which can divide into naught butunknotitselfsaidbe prime.

The product ofAlexander polynomialstwo knotsan Alexander polynomialtheir sum. Seeing thatgranny knot issumtwo trefoils ofsame hand, andsquare knot issumtwo trefoilsopposite hand, we can easily calculate their polynomial. (They sharepolynomial sincehandedness oftrefoilnot detected.)

granny square

Ref: Mark Anthony Armstrong Basic Topology (Springer-Verlag 1987) p237–9

Note: Because ofMathworld form, I suspect Alexander polynomials havecoefficient symmetry which leads tosecond canonic form. The polynomial above will have degree 2n; divide by xⁿcollect xⁱx^-i terms. Eg, trefoil: figure-eight: granny/square: stevedore:

• this seemsbecopyMathworld
• ditto
• second Alexander polynomial

See skein relations forsecond waycompute Alexander polynomials.

Alexander-Conway polynomial

Even before Conway foundskein-relation approach toAlexander polynomials,second form via changevariable was apparent. But Conway getscredit.

This other polynomialusually denoted forlink (generalised knot) L. Its skein-relation equation is

with

It relates tonormalised Alexander polynomial as

• Ref Mathworld

Jones polynomial

In 1984 Vaughn F. R. Jones came out withfirst really new knot polynomial since Alexander's. He was tinkeringhis specialty, von Neumann algebras,almost by accident found this linkageknot theory. (Knot theory beganan idea that atoms were knotted æther vortices,von Neumann algebraskeyquantum theory,successoratomic study. Jones' discovery was thussortfamily reunion.)

In skein relation

with .

Can sometimes distinguishknot from its reflection; this isgreat "breakthrough" overAlexanderConway polynomials.

where L isreflection.

all knots K

all links L

• Ref Mathworld

HOMFLY(PT) polynomial

Jones' discovery promptedhunt forstructure above his polynomialAlexander's. Five collaborations found one essentially simultaneously; four published jointly1985 rather than fight over priority. "HOMFLY"derived from their initials: Jim Hoste, Adrian Ocneanu, Kenneth C. Millett, Peter J. Freyd, W. B. Raymond Lickorish,David N. Yetter. Some authors write "HOMFLYPT"includepairPoles, Józef H. Przytycki" class="external">http://home.gwu.edu/~przytyck/-->Pawel Traczyk, who got left out dueslow mail service.

HOMFLYPT isbinary (two-variable) polynomial, as withpredecessors. But three different skein relations (and thus three slightly different polynomials)seen inwild:

(Doll & Hoste 1991, Kanenobu & Sumi 1993)

(Kauffman 1991)

(Lickorish & Millett 1988)

For maximal confusion therealsoternary form

Forlink Ln unlinked unknots,commonin skein recurrences, iteasily shown (by induction) that

The simplicity ofternary HOMFLYPTdeceptive;actually encapsulatessignificant classknot functions. Given any three functions Q, R, S (oversame set intofield),skein-relation equation

is satisfied by . This obviously includesAlexander, Conway,Jones polynomials:

Thus,go any furtherskein relations one must avoid recurrences ofabove form.

Such interrelations permit facts about HOMFLYPTbe transferred (with appropriate transformation)its predecessors. For instance, although the cinquefoilimage::Knot-10-132-sm.pngknownbe different knots, their HOMFLYPTs aresame; thusalso share their Alexander, Conway,Jones. (Worse, two 10-crossing knots, image:knot-10-25-sm.pngimage:knot-10-56-sm.png,insame boat; thus itnot helpfulpair polynomialcrossings.)

Also, all knot sums —andother polynomials inherit this property.

<The authorastounded thatternary HOMFLYPT, which seems an absurdly obvious skein relation, should have lain unseenplain sightover 20 years. Conway must really be wondering why he didn't see it. Perhaps he thoughtwas too obviouswork.>

<The authoralso puzzled that Mathworld mentionsternary onHOMFLYPT page as ifwereHOMFLYPT, but without specific citation,doesn't useform anywhere else—very odd, given that it'sform from which six other polynomialsreadily found.>

• Ivars Peterson Mathematical Tourist (1988) p70–80
• Mathworld
• Calculating HOMFLY by Dynamic Programming

BLM/Ho polynomial

• Brandt, Lickorish, Millett, Ho
• ''involvesskein rel eqnan ... WTF? I mean, there's an obvious 4th waypatching, but notdirections...isundirected? At first glance BLM/Ho's skein recurrenceproperly symmetricitwork... And
• Ref Mathworld

Kauffman unary polynomial

Louis H. Kauffman has two knot polynomialshis credit. Also known as normalised bracket polynomial. Denoted by by Kauffman but other authors have used different letters. Itvery likeJones polynomial:

• Ref Mathworld

Kauffman binary polynomial

It isgeneralisation ofJones polynomial

but other than having more terms thanHOMFLYPT polynomial, its relation tolatterunknown.

It relatesKauffman's unary polynomial as

• Ref Mathworld

Unworked examples

knot K	Alexander	Conway	Jones
unknot image:knot-unknot-64.png	1	1	1
left trefoil image:knot-trefoil-left-64.png
right trefoil image:knot-trefoil-right-64.png
(right?) cinquefoil image:knot-cinquefoil-sm.png
figure-8 image:knot-figure8-64.png
square image:knot-square-64.png
(left?) granny image:knot-granny-64.png
stevedore image:knot-stevedore-sm.png

(Composing notes)

• Saw mention ofMillet-Prztycki-Traczyk polynomial by three ofHOMFLYPTers...
• WTF Siefert matrices?
• [1]
• might be usefulsomething
• HOMFLY this!