Pis'ma v ZhETF, vol. 101, iss. 12, pp. 931-934
© 2015 June 25
On the defect and stability of differential expansion
Ya. Kononov+, A. Morozov^ Institute for Theoretical and Experimental Physics, 117218 Moscow, Russia
National Research Nuclear University "MEPhI", 15409 Moscow 1, Russia
Institute for Information Transmission Problems, 127994 Moscow, Russia
+Higher School of Economics, Math Department, 117312 Moscow, Russia
Submitted 30 April 2015
Empirical analysis ol many colored knot polynomials, made possible by recent computational advances in Chern-Simons theory, reveals their stability: for any given negative N and any given knot the set ol coefficients ol the polynomial in r-th symmetric representation does not change with r, il it is large enough. This lact reflects the non-trivial and previously unknown properties ol the differential expansion, and it turns out that from this point of view there are universality classes of knots, characterized by a single integer, which we call defect, and which is in fact related to the power of Alexander polynomial.
DOI: 10.7868/S0370274X15120115
HOMFLY polynomials are Wilson-loop averages in 3d Chern-Simons theory [1], which in this simplest model depend only on the topology of the Wilson line (knot). Therefore one can separate and study the group-theory properties of observables - and this is a nontrivial and very interesting problem, for a brief summary of results see [2]. From the quantum field theory perspective knot polynomials are direct generalization of conformal blocks, and this relation [3] provides one of the effective calculational methods in knot theory.
Recent advances in [4, 5], based on the previous considerations in [3-15], provided a way to systematically calculate simplest colored HOMFLY polynomials [16] for a really wide variety of knots - including, in particular, the entire Rolfsen table of [17]. This allows us to return to the study of "differential expansions" of [1827], which was temporarily postponed because of the insufficient "experimental" material.
In this note we describe empirically obtained properties of these expansions for symmetric representations [r] (where r is the length of the single-line Young diagram). It looks like there are different universality classes of such expansions, characterized by a single integer, which we call "defect" S,c. Moreover, these newly observed properties allow to identify 2(S,c + 1) with the power of Alexander polynomial and lead to a peculiar stability property of symmetrically colored HOMFLY for large enough r: what stabilizes is not the polynomial itself, but the set of its coefficients - i.e. something
like the "coordinates" grj, introduced in [24]. Theoretical analysis of these observations, proofs and extension to non-(anti)symmetric representations are beyond the scope of the present text.
1. The notion of defect. Differential expansion provides a knot-dependent (/-deformation (quantization) of the remarkable factorization property [9,11,19-21] of colored "special" polynomials at q = 1,
H%(A) = (H[^A))1
V
9=1
V representation R and knot KL
(1)
which fully defines their dependence on representation (Young diagram) R. Currently these expansions can be well studied only for symmetrically-colored HOMFLY, and we focus on this case in the present paper. The story starts from the fact that • Hr = H[r] always possesses differential expansion of the following form:
q2) = 1 +
s — 1
=i
3=0
-^e-mail: morozov@itep.ru
For generic knot Gf is a non-factorizable Laurent polynomial of A and q, but for some knots it can be further factorized. In this formula we use the notation {x} = x — x^1 and quantum number is defined as
N = Un}/Uh
• what is important, if Gf is divisible by some "differential" {Aqk}, the same is true for all other Gf, with
IhicbMa b >K3TO tom 101 Bbin. 11-12 2015
931
s' > s. This property allows one to introduce defect
functions vf and = s — 1 — vf :
"f-i
Gf(A,q) = F^A,q) J] {V}
j=o
= F?(A,q) H {V}
3=0
(3)
which are both(!) monotonically increasing function of s,
< < /;> for s < s'
(4)
i.e. both grow - but not faster than s;
• for A = qN with any fixed N, positive or negative,
Fs(qN,q)~{q}'i- Gs{q , q) ~ {q}
1 s-l
(5)
i.e. at fixed N the s-the term of differential expansion is actually of the order {q}2s;
• it turns out that vf as a function of s has a very special shape, fully parameterized by a single integer S,c > —1, which we call the defect of differential expansion:
defect S,c = -1
(6)
Immediately observable are two remarkable properties of this kind:
• Gf{A,q) has power 2SKj in q2, i.e. Gf{A,q) =
= E ctfi-,
3 = SK
For example, S,c = 0 whenever Gf is independent of q;
• Alexander polynomial has power 2(S,c + 1) in q2,
i.e.Al'c(q) = H[c(A=l,q)= £ ajq
j=-SK--1
1,
■2j
5f~ = -Powerg2 (Ar) - 1.
(12)
For S,c ^ 0 these facts are not immediately related: contributing to Alexander polynomials are all Gf with s < S,c + 1 and they can and do contain much higher powers in q. Moreover, even the product of differentials in the s-term has power in q, which grows quadratically with s - and thus with S,c. This means that there are serious cancelations behind the linear law (12).
Since Alexander polynomials are easily available already from [17], the values of /x, v for each knot are easily obtained from this data.
3. Twist and torus knots. For all twist knots the defect is vanishing
^twist _ Q
(13)
defect S,c = 0
defect ô^ = 1
defect S
■ic
■ic
defect S
In general
vf = entier
1^=8-1,
ns ~ -, vs = entier
2 s ir-Ms ~ y, vs = entier
s - 1
s - 1
r 3s r is — 1
~ y, vs = entier I ——
(7)
,(8)
,(9)
,(10)
Instead for torus knots it is a kind of maximal:
for the 2-strand family ¿I2'™! = for the 3-strand family (819, 10124, •••) for the 4-strand family
n — 3
n - 2, 3n — 5
in general,
¿1"
mn — m — n — 1
/; = s-l- vf
s - 1
s^Ti J ~ s^ + v
6>c
(11)
6,c +1 '
2. Relation to Alexander polynomial. It is an
interesting question, if the value of S,c can also restrict the coefficient functions Ff (A, q).
(14)
since the power of Alexander polynomial IS
(to — l)(n — 1).
4. Negative defect: KTC mutants and their relatives. Starting from 11 intersections there are cases when Alexander is just unity, i.e. the defect is negative, S,c = —1. According to our general rules this means that for such knots already Gf is reducible: Gf ~ {A}. Of course, also all other Gf ~ {A}, because all the terms of the differential expansion are vanishing for A = 1.
This is indeed true for the first example - the celebrated Kinoshita-Terasaka and Conway (KTC) mutants K, = lln42 & lln34, reconsidered recently in [5], - and also for the next example, available from [17]: tC = 12n313 & 12n430. Moreover, the combination of
On the defect and stability of differential expansion
933
[4] and [5] allows to calculate HOMFLY for KTC mutants for any symmetric representation and validate (6) in this particular example.
5. Summary: stability and other properties of differential expansion. Take any randomly chosen knot (say, K. = 62).
It is easy to observe that, starting from H^2, the sets of coefficients are the same - despite the polynomials are different. At A = q~2 the same happens, beginning from i/f2. Thus what stabilizes are not the polynomials themselves, but something else, more appropriately associated with the knots. In full accordance with the vision in [24] this something are the coefficient functions Gf of the differential expansion.
Due to their properties, which are revealed in the present paper, contributing at A = q~N are just the first few terms of the expansion (2):
H- A
(N+1)(SK + 1)
[N+l]Gf(A = q
M
s-iM!
1- E
s=0 s — 1
x J]{<f-N+j}{<f-jh
0=0
-N\
(15)
where the last product is Laurent polynomial in qr and due to (5) the ratio in front of it is an r-independent polynomial. Thus what we get is just a sum of a few polynomials, multiplied by different powers of qr. They do not overlap at large enough r, and this provides an r-independent set of the coefficients, as in the above example.
In fact, one could wish to interpret the remarkable identity [19, 20]
H^{A=l,q) = Hf{A=l,qr
(16)
for Alexander polynomials clS cl manifestation of the same phenomenon at N = 0. However this is literally so only for ô,c = —1 and S,c = 0. Still (16) is true not only for all knots, but actually for all single-hook (and not just single-line) representations R. For such representations (16) is a kind of a dual to (1).
6. Conclusion. In this paper we studied the "quality" of the differential expansion (2) for symmetrically colored reduced HOMFLY polynomials - the typical observables in the simplest possible Yang-Mills theory. If only naive representation-theory properties are taken into account from (1) to restriction I < N on the number I of lines in the Young diagram for particular SL(N), this expansion has the form (2) with irreducible polynomial coefficient functions Gs(A,q). It
is well known, however, that sometime Gs are further factorized, thus adding more restrictions/structures to the color-dependence of physical observables. Now, when methods were developed to study entire classes of generic knots, we could attack this problem in a systematic way - and demonstrate that Gs are always factor-izable for high enough s. The depth of factorization appeared to depend on a single characteristic of the knot, which we originally called defect of the expansion, and further demonstrated that it is linearly related to the degree of Alexander polynomial, what makes it very easy to find.
This factorization universality leads to remarkable kind of stabilization of symmetrically colored HOMFLY - ensuring that increasing r beyond some knot-dependent boundary does not provide new physical (topological) information. This is what one naturally expects, and now we see how this actually works.
Highly desirable is extension of this new insight beyond pu
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