Sequences of knots and their limits

0.04 MB·English

by  Pedro Lopes

Sequences of Knots and Their Limits
Pedro Lopes
DepartmentofMathematics,InstitutoSuperiorTécnico,TechnicalUniversityofLisbon,
8 Av.RoviscoPais,1049-001LisbonPortugal
0
0
2 Abstract. Hyperfiniteknots,orlimitsofequivalenceclassesofknotsinducedbyaknotinvariant
takingvaluesinametricspace,wereintroducedinapreviousarticlebytheauthor.Inthisarticle,
n
wepresentnewexamplesofhyperfiniteknotsstemmingfromsequencesoftorusknots.
a
J Keywords: Knots,Hyperfiniteknots,quandles,colorings,CJKLSinvariant

5 PACS: 02.10.Kn
2

]
T INTRODUCTION
G
. Consider a sequence of knots of increasing crossing number, where crossing number
h
t of a knot stands for the least number of crossings it takes to draw a diagram of the
a
m correspondingknot.Doesitmakesensetodiscussthelimitofsuchasequence?Itdoes,
to a certain extent. Assume there is a knot invariant, say f, which takes on values in
[
a complete metric space, call it M. Define two knots to be related if and only if they
1
have the same f-invariant. Let K be the set of the equivalence classes of the knots
v f
1 under this (equivalence) relation. There is then an f∼ map from K to M which is
f
0 obviously injective. Hence, K can be regarded as a metric subspace of M. Let then
9 f
3 K betheclosureofK inthetopologyofM.Inthisway,wheneverthesequenceof f-
f f
1. invariantsofagivensequenceofknotsconvergesinM,thenthecorrespondingsequence
0 ofequivalenceclassesoftheknotsconvergesinK .Wecallthelatterlimitahyperfinite
f
8
knot([1]).
0
:
v
One way of implementing these ideas is via the CJKLS invariant of knots ([1, 2]).
i
X There is a CJKLS invariant for each choice of labeling quandle X, abelian group A,
r and 2-co-cycle f in H2(X,A). Furthermore, the CJKLS invariant is a sum-over-states,
a
where the states are the colorings of the knot under study by the labeling quandle X,
and thefor each coloring thef ’s are evaluatedat each crossingoftheknotdiagram and
then multiplied.Since the CJKLS invariant takes on values on the group-algebra of the
abeliangroupA,weembeditintheeuclideanspaceR|A|,therebyobtaininganinvariant
takingonvaluesinacompletemetricspace.Wethentakelogarithmsofeachcoordinate,
obtaining what we call the free energy invariant, and then by dividing by the crossing
number we obtain what we call the free energy per crossing. We were clearly inspired
byThermodynamicsand by StatisticalMechanics ofExactlySolvedModels([3, 4]).
HYPERFINITE KNOTS WHOSE FREE ENERGIES PER
CROSSING EQUAL THE NULL VECTOR
Specifically, a quandle is an algebraic structure i.e. a set, X, endowed with a binary
operation,∗,such thatforall a,b,c∈X:
• a∗a=a;
• thereisauniquex∈X s.t.x∗b=a;
• (a∗b)∗c=(a∗c)∗(b∗c)
Any group with conjugation as the ∗ operation is a quandle. The ring of Laurent
polynomials on the variable T, with integer coefficients, Z[T,T−1], endowed with the
operation a∗b=Ta+(1−T)b is also a quandle. Quotients obtained by factoring this
quandle by certain ideals in the ring of Laurent polynomials are finite quandles called
Alexander quandles. Any knot (oriented) diagram provides a quandle by regarding the
arcs of the diagram as generators and reading relations at the crossings of the sort
under-arc ∗ over-arc equals under-arc where the co-orientation at the over-arc points
to the under-arc that receives the product. Moreover this quandle is invariant under the
Reidemeister moves thus constituting a knot invariant called the knot quandle. Upon
fixingafinitequandleX thenumberofhomomorphisms(alsoknownascolorings)from
theknotquandletothefinitequandleX isalsoaknotinvariant.Thisinvariantisreferred
to as counting colorings. An assignement of elements of X (colors) to the knot quandle
is a homomorphism if at the crossings the colors assigned to the arcs satisfy the same
relationsasthearcsdo.Oneefectivewayoffindingcoloringsofaknotbyagivenfinite
quandleX resortstobraids([6]).Anyknotcanberepresentedbytheclosureofacertain
braid so we assign colors to the strands at the top of the braid (the input colors) whose
closure yields the knot under study. These colors propagate down the braid giving rise
to knew colors as one goes past each crossing and eventually obtaining a new string of
colors on the strands at the bottom of the braid (the outputcolors). If this assignmentis
a coloring then the string of colors at the top equals the string of colors at the bottom,
otherwiseitisnot acoloring.Insomeway,thebraidacts ontheinputcolorsto produce
the output colors; if the output colors match the input colors then these input colors
stand for a coloring, otherwise they do not stand for a coloring. We will assume that
the finite labeling quandles we will be using are all Alexander quandles. Furthermore,
we use the Burau representation ([6]) to perform the calculations. The braid group on
N strands, B , is generated by the so-called standard generators, s , i = 1,...,N−1
N i
([6]). Geometrically, s is the (i+1)-th strand going over the i-th strand. In the Burau
i
representation the standard generators are mapped to matrices over the ring of Laurent
polynomialsin thevariableT as follows:
0 T
s 7−→I ⊗ ⊗I
i i−1 1 1−T N−1−i
(cid:20) (cid:21)
Then the action of the braid on the input color plus the matching against the input we
referred to,translatesintothe“eigenvector”equation
(a a ...a )B(b)=(a a ...a )
1 2 N 1 2 N
wherea ,i=1,...,N arethecolorsfromtheAlexanderquandleX andB(b)istheBurau
i
e e
matrix of the braid b=s i1 ·····s iL whose closure yields the knot under study. If the
i i
1 L
equality holds then (a ,...,a ) stands for a coloring, otherwise it does not stand for a
1 N
coloring.Moreover,notethattheequalityisunderstoodinthequotientcorrespondingto
theAlexanderquandlewearedealing with.
As sets, these Alexander quandles are finite. Moreover, they have a ring structure
induced by addition and multiplication of polynomials. For a given positive integer N,
the ring of N×N matrices with entries from one of these Alexander quandles is thus
alsofinite.SinceeachBuraumatrix,sayB,isinvertible,thenithasfiniteorderi.e.,there
isa(positive)integerM suchthat BM istheidentitymatrix.
Given a knot K and a braid b∈B whose closure yields K , consider the sequence
1 N 1
of knots given by K =bn, where ... denotes “closure”. When coloring the knots from
n
this sequence with an Alexander quandle, we write down the “eigenvector” equation as
above: b
c
n
(a a ...a ) B(b) =(a a ...a )
1 2 N 1 2 N
for knot K and where B(b) is the Burau matrix for b in the appropriate quotient. Since
n (cid:0) (cid:1)
there is an M such that B(b)M is the identity matrix, then there is periodicity in the
colorings of the knots of this sequence, in particular, in the sense that there are only M
distinct sets of eigenvectors for this sequence of knots. Moreover there are at most M
distinct distributions of colors per arcs over any M consecutive factors of bn. Also, for
a fixed bn, suppose n = lM+r; there are l identical coloring patterns followed by the
coloringpattern associatedto thebr.
This periodicity issue was first discussed in [5] if only for some sequences of torus
knots. We are now claiming it in more generality, both for all Alexander quandles and
notonlyfortheparticularcase ofsequences oftorusknots.
NowassumethatanabeliangroupAanda2-co-cyclef havebeenchosenbesidesthe
Alexander quandle X. Let |A| stand for the cardinality of A and let M be as above i.e.,
B(b)M istheidentitymatrix.Givenapositiveintegern let
n=lM|A|+r with0≤r<M|A|
Weletc(bm)denotethesetofcrossingsofbm and1 denotetheidentityinthegroupA.
A
TheCJKLS invariantofK =bn thenreads
n
l|A|
Z(bn)= (cid:229) (cid:213) f etb= (cid:229) (cid:213) f et · (cid:213) f et =
a1,...,aN∈Xt ∈c(bn) a1,...,aN∈X t ∈c(bM) ! t ∈c(br)
s.t.... s.t....
b
|A| l
= (cid:229) (cid:213) f et · (cid:213) f et = (cid:229) 1 l· (cid:213) f et =
A
a1,...,aN∈X t ∈c(bM) ! ! t ∈c(br) a1,...,aN∈X t ∈c(br)
s.t.... s.t.... (cid:0) (cid:1)
= (cid:229) (cid:213) f et
a1,...,aN∈Xt ∈c(br)
s.t....
The dots next to s.t. and under a ,...,a ∈ X stand for the conditions the a have
1 N i
to satisfy in order to stand for colorings; these conditions stem from the eigenvector
equation and there are at most M distinctconditions.Since 0≤r <M|A| then there are
atmostM|A|differentvaluesoftheCJKLSinvariantforthissequenceofknotsK .Since
n
we regard the CJKLS invariant as taking on values in the euclidean metric space R|A|
then the sequence of the CJKLS invariant for K is bounded and so is the sequence of
n
thefreeenergiesforthissequenceofknots.IfthecrossingnumberofK goestoinfinity
n
as n goes to infinity then the free energy per crossing of this sequence of knots goes to
thenullvector.
EXAMPLE
FixapositiveintegerN andconsiderthesequenceoftorusknots T(N,n) .Wehave
n∈N
n (cid:0) (cid:1)
T(N,n)= s s ···s s
N−1 N−2 2 1
(cid:18) (cid:19)
MoreoverthecrossingnumberofT(N,n)is
min{|N|(|n|−1),|n|(|N|−1)}
([7])whichtendstoinfinityasntendstoinfinity.Thus,thesequenceofthefreeenergies
percrossingfromaCJKLSinvariantwithanAlexanderquandleaslabelingquandle,for
asequenceoftorusknotsastheoneindicatedtendstothenullvectortherebygivingrise
toahyperfiniteknot.
ACKNOWLEDGMENTS
TheauthoracknowledgesfinantialsupportfromPOCTI/FCT/FEDER.
REFERENCES
1. P.Lopes,Chaos,SolitonsandFractals,341450–1472(2007).
2. J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford,M. Saito, Trans. Amer. Math. Soc., 355 3947-
3989(2003)
3. H.B.Callen,Thermodynamicsandanintroductiontothermostatistics,2ndedition,JohnWileyand
Sons,NewYork1985
4. R.J.Baxter,Exactlysolvedmodelsinstatisticalmechanics,AcademicPress,NewYork1982
5. J.S.Carter,D.Jelsovsky,S.Kamada,M.Saito,Adv.Math.,15736-94(2001)
6. J.Birman,Braids,links,andmappingclassgroups,AnnalsofMath.Studies82,PrincetonUniversity
Press,Princeton,N.J.,1974
7. K.Murasugi,Trans.Amer.Math.Soc.,326237-260(1991)