On Chebyshev Polynomials and Torus Knots
In this work, we demonstrate that the q-numbers and their two-parameter generalization, the q,p -numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely connected with the Chebyshev polynomials, can also be related wit...
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nasplib_isofts_kiev_ua-123456789-132952025-02-09T22:47:55Z On Chebyshev Polynomials and Torus Knots Про поліноми Чебишова і торичні вузли Gavrilik, A.M. Pavlyuk, A.M. Загальні питання теоретичної фізики In this work, we demonstrate that the q-numbers and their two-parameter generalization, the q,p -numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely connected with the Chebyshev polynomials, can also be related with the Alexander polynomials for the class T(s, 2) of torus knots, s being an odd integer, and used for finding the corresponding skein relation. Then, we develop this procedure in order to obtain, with the help of q, p - numbers, the generalized two-variable Alexander polynomials and to prove their direct connection with the HOMFLY polynomials and the skein relation of the latter. У роботi показано, що q-числа та їх двопараметричнi узагальнення, q, p-числа можна використати для отримання деяких полiномiальних iнварiантiв торичних вузлiв i зачеплень. По-перше, показано, що q-числа, якi тiсно пов’язанi з полiномами Чебишова, можуть бути пов’язанi з полiномами Александера для класу T(s, 2) торичних вузлiв, де s – непарне цiле число, i використанi для знаходження вiдповiдного скейн-спiввiдношення. Потiм використано цю процедуру для отримання за допомогою q, p-чисел, двопараметричних узагальнених полiномiв Александера та показано зв’язок останнiх iз полiномiальними iнварiантами HOMFLY та їх скейн-спiввiдношенням. This research was partially supported by the Grant 29.1/028 of the State Foundation of Fundamental Research of Ukraine and by the Special Program of the Division of Physics and Astronomy of the NAS of Ukraine. 2010 Article On Chebyshev Polynomials and Torus Knots / A.M. Gavrilik, A.M. Pavlyuk // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 129-134. — Бібліогр.: 21 назв. — англ. 2071-0194 PACS 02.10.Kn,02.20.Uw https://nasplib.isofts.kiev.ua/handle/123456789/13295 en application/pdf Відділення фізики і астрономії НАН України |
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Загальні питання теоретичної фізики Загальні питання теоретичної фізики |
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Загальні питання теоретичної фізики Загальні питання теоретичної фізики Gavrilik, A.M. Pavlyuk, A.M. On Chebyshev Polynomials and Torus Knots |
| description |
In this work, we demonstrate that the q-numbers and their two-parameter generalization, the q,p -numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely connected with the Chebyshev polynomials, can also be related with the Alexander polynomials for the class T(s, 2) of torus knots, s being an odd integer, and used for finding the corresponding skein relation. Then, we develop this procedure in order to obtain, with the help of q, p - numbers, the generalized two-variable Alexander polynomials and to prove their direct connection with the HOMFLY polynomials and the skein relation of the latter. |
| format |
Article |
| author |
Gavrilik, A.M. Pavlyuk, A.M. |
| author_facet |
Gavrilik, A.M. Pavlyuk, A.M. |
| author_sort |
Gavrilik, A.M. |
| title |
On Chebyshev Polynomials and Torus Knots |
| title_short |
On Chebyshev Polynomials and Torus Knots |
| title_full |
On Chebyshev Polynomials and Torus Knots |
| title_fullStr |
On Chebyshev Polynomials and Torus Knots |
| title_full_unstemmed |
On Chebyshev Polynomials and Torus Knots |
| title_sort |
on chebyshev polynomials and torus knots |
| publisher |
Відділення фізики і астрономії НАН України |
| publishDate |
2010 |
| topic_facet |
Загальні питання теоретичної фізики |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/13295 |
| citation_txt |
On Chebyshev Polynomials and Torus Knots / A.M. Gavrilik, A.M. Pavlyuk // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 129-134. — Бібліогр.: 21 назв. — англ. |
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2025-12-01T13:24:03Z |
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| fulltext |
ON CHEBYSHEV POLYNOMIALS AND TORUS KNOTS
ON CHEBYSHEV POLYNOMIALS AND TORUS KNOTS
A.M. GAVRILIK, A.M. PAVLYUK
Bogolyubov Institute for Theoretical Physics, Nat. Acad. Sci. of Ukraine
(14b, Metrolohichna Str., Kyiv 03680, Ukraine; e-mail: omgavr@bitp.kiev.ua; pavlyuk@bitp.kiev.ua)
PACS 02.10.Kn,02.20.Uw
c©2010
In this work, we demonstrate that the q-numbers and their two-
parameter generalization, the q,p -numbers, can be used to obtain
some polynomial invariants for torus knots and links. First, we
show that the q-numbers, which are closely connected with the
Chebyshev polynomials, can also be related with the Alexander
polynomials for the class T (s, 2) of torus knots, s being an odd in-
teger, and used for finding the corresponding skein relation. Then,
we develop this procedure in order to obtain, with the help of q, p -
numbers, the generalized two-variable Alexander polynomials and
to prove their direct connection with the HOMFLY polynomials
and the skein relation of the latter.
1. Introduction
The relevance of knots and links to many physical [1–
3] and biophysical [4] systems implies the importance of
investigating the properties and characteristics of knot-
like structures. The concepts of knot theory play the
important role in the models of statistical physics [5],
quantum field theory [6], quantum gravity [7], and in a
number of other physical phenomena. In the preprint
of 1975, it was proposed by L.D. Faddeev that knot-like
solitons could be realized in a nonlinear field theory [8],
in a definite model defined in 3 + 1 dimensions. The
model includes the standard nonlinear O(3) σ-model,
which admits static solitons in 2 + 1 dimensions, and a
Skyrme term. In the Faddeev model, the static solitons
are stabilized by the integer-valued Hopf charge. Inter-
est in the model was renewed in 1997 after the article
of Faddeev and Niemi in [9]. They have made first at-
tempts at a numerical construction of solitons with the
minimal energy in the form of knots. Battye and Sut-
cliffe demonstrated that, for a higher Hopf charge, the
twisted, knotted, and linked configurations occur [10]. In
particular, they showed that the minimal energy soliton
with Hopf charge seven is a trefoil knot.
R.J. Finkelstein has proposed a field theory model, in
which the local SU(2) × U(1) symmetry group of the
standard electroweak theory is combined with the global
quantum group SUq(2), the symmetry group of knotted
solitons [11, 12]. This allows one to incorporate a q-
soliton into field theory and to replace point particles by
knotted solitons. The more recent discussion on the role
of field theory knots both in superconductivity theory
and in the Yang-Mills theory can be found in [13].
In the context of modeling the static properties of
hadrons, it was shown in [14] (see also [15]) that global
quantum groups SUq(n), n = 2, . . . , 6, can be success-
fully applied to flavor symmetries, and certain torus
knots put into correspondence, through Alexander poly-
nomials, with vector quarkonia.
Various polynomial invariants are known to be one of
the basic characteristics of knots and links (see e.g. [16]).
Among them, the Alexander polynomials, Jones polyno-
mials, and HOMFLY polynomials are the best studied
and play the important role in the knot theory and its
applications.
To describe some properties and characteristics of
knots and links, the classical Chebyshev polynomials can
be used. For example, in [17], the Chebyshev polynomi-
als were utilized for the polynomial parametrization of
noncompact counterparts of torus knots. It was shown
how to construct the Chebyshev model associated with
any knot.
In this paper, we concentrate on studying the poly-
nomial invariants such as the Alexander and HOMFLY
polynomials and their close connection with the Cheby-
shev polynomials. We restrict ourselves to the set of
torus knots and links and show that a certain rather sim-
ple two-variable generalization of the Chebyshev polyno-
mials is well suited for characterizing those knots.
2. Alexander Polynomials and a Skein Relation
The Alexander polynomials A(t) for knots and links can
be defined (see e.g. [16]) by the skein relation
A+(t) = (t
1
2 − t− 1
2 )AO(t) +A−(t) (1)
and the condition for the unknot:
Aunknot = 1. (2)
Using (1) and (2), one can find the Alexander polynomial
for any knot or link, by applying to it, in a standard way,
the surgery operations of switching and elimination.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 129
A.M. GAVRILIK, A.M. PAVLYUK
From now on, we consider torus knots and links of the
type (s, 2), where s is any positive integer. If s = 1, we
have the unknot, the case of s = 2 corresponds to the
Hopf link, s = 3 to the trefoil knot, and so on. In general,
when s is odd, we have the series of torus knots T (s, 2),
and if s is even, we have the series of two-component
torus links L(s, 2). Here, s equals the minimal number
of crossings.
Applying the operation of elimination to (s, 2), one
obtains (s−1, 2), whereas the switching operation turns
(s, 2) into (s − 2, 2), for s > 2. This means that
A+(t), AO(t) and A−(t) correspond to three successive
Alexander polynomials, which allows one to make the
following juxtaposition in (1):
A+(t)→ Ã2
n+1(t), AO(t)→ Ã2
n(t),
A−(t)→ Ã2
n−1(t).
(3)
Thus, from (1) and (3), one obtains the recurrence re-
lation for the tilded Alexander polynomials for a unified
set of torus knots and links of the type (s, 2), (the poly-
nomials are arranged by increasing degrees):
Ã2
n+1(t) = (t
1
2 − t− 1
2 )Ã2
n(t) + Ã2
n−1(t). (4)
It is convenient to denote the Alexander polynomials
for a subset containing only torus knots (or the subset
of torus links) as
As,2m (t) ≡ As,2m ≡ A2
m.
Here, m is the degree of the corresponding Alexander
polynomial, which has the form of a Laurent polynomial:
m =
1
2
(s− 1). (5)
Since s = 2m + 1 for both knots and links, the degree
m of the Alexander polynomial for knots T (s, 2) is an
integer, and m for links L(s, 2) is half-integer. Let us
first give the table of the Alexander polynomials As,2m (t)
for torus knots T (s, 2) ≡ T (2m+ 1, 2)
A1,2
0 (t) = 1,
A3,2
1 (t) = t− 1 + t−1,
A5,2
2 (t) = t2 − t+ 1− t−1 + t−2,
A7,2
3 (t) = t3 − t2 + t− 1 + t−1 − t−2 + t−3,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
A2m+1,2
m (t) = tm − tm−1 + · · · − t−(m−1) + t−m =
=
m∑
i=0
tm−2i −
m−1∑
i=0
tm−2i−1 =
2m∑
i=0
(−1)itm−i.
(6)
The recurrence formula for polynomials (6) looks as
(dropping 2m+ 1 in superscript)
A2
m+1(t) = (t+ t−1)A2
m(t)−A2
m−1(t). (7)
Now consider torus links of the type L(s, 2) ≡ L(2m+
1, 2), where s is a positive even integer. The degree m
of the Alexander polynomial As,2m (q) is again as in (5).
It is half-integer now. The table shows the Alexander
polynomials for these torus links:
A2,2
1
2
(t) = t
1
2 − t− 1
2 ,
A4,2
3
2
(t) = t
3
2 − t 1
2 + t−
1
2 − t− 3
2 ,
A6,2
5
2
(t) = t
5
2 − t 3
2 + t
1
2 − t− 1
2 + t−
3
2 − t− 5
2 ,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
A2m+1,2
m (t) = tm − tm−1 + · · ·+ t−(m−1) − t−m =
=
2m∑
i=0
(−1)itm−i.
(8)
Note that polynomials (8) satisfy the recurrence relation
(7) too, though now with half-integer subscripts. So, for
torus knots and links of the type (s, 2), Eq. (7) describes
either three successive torus knots (if s is an odd integer)
or three successive torus links (if s is an even integer).
Unifying two tables (6) and (8), we have the table of
the Alexander polynomials for torus knots T (s, 2) and
links L(s, 2), where s = 2m + 1 is an integer, while m
is an integer or half-integer and equals the degree of the
Alexander polynomial:
Ã1,2
0 (t) ≡ A1,2
0 (t) = 1,
Ã2,2
1 (t) ≡ A2,2
1
2
(t) = t
1
2 − t− 1
2 ,
Ã3,2
2 (t) ≡ A3,2
1 (t) = t− 1 + t−1,
Ã4,2
3 (t) ≡ A4,2
3
2
(t) = t
3
2 − t 1
2 + t−
1
2 − t− 3
2 ,
Ã5,2
4 (t) ≡ A5,2
2 (t) = t2 − t+ 1− t−1 + t−2,
Ã6,2
5 (t) ≡ A6,2
5
2
(t) =
= t
5
2 − t 3
2 + t
1
2 − t− 1
2 + t−
3
2 − t− 5
2 ,
Ã7,2
6 (t) ≡ A7,2
3 (t) =
= t3 − t2 + t− 1 + t−1 − t−2 + t−3,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
Ã2m+1,2
2m (t) ≡ A2m+1,2
m (t) =
= tm − tm−1 + tm−2 − · · · t−m =
2m∑
i=0
(−1)itm−i.
(9)
130 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
ON CHEBYSHEV POLYNOMIALS AND TORUS KNOTS
Two different notations for the Alexander polynomials
are related to each other as
Ã2m+1,2
2m (t) = A2m+1,2
m (t).
The recurrence relation for polynomials (9) is given by
(4) which immediately follows from the skein relation (1)
in view of correspondence (3).
Let us show that (4) can be obtained from (7) as well.
To see that, we first write (1) in the general form
A+(t) = b1AO(t) + b2A−(t). (10)
With account of (3), we also have the recursion relation
Ã2
n+1(t) = b1Ã
2
n(t) + b2Ã
2
n−1(t). (11)
Then we rewrite Eq. (7) in terms of tilded Ã2
n(t):
Ã2
n+1(t) = (t+ t−1)Ã2
n−1(t)− Ã2
n−3(t). (12)
The latter in general terms looks as
Ã2
n+1(t) = c1Ã
2
n−1(t) + c2Ã
2
n−3(t). (13)
From (11) we have
Ã2
n(t) = b1Ã
2
n−1(t) + b2Ã
2
n−2(t), (14)
Ã2
n−1(t) = b1Ã
2
n−2(t) + b2Ã
2
n−3(t). (15)
Insert (14) into (11):
Ã2
n+1(t) = (b21 + b2)Ã2
n−1(t) + b1b2Ã
2
n−2(t). (16)
Then we put Ã2
n−2(t) from (15) into (16)
Ã2
n+1(t) = (b21 + 2b2)Ã2
n−1(t)− b22Ã2
n−3(t). (17)
The comparison of (17) and (13) gives
c1 = b21 + 2b2, c2 = −b22. (18)
From Eq. (18), we have
b1 = (c1 − 2b2)
1
2 , b2 = (−c2)
1
2 . (19)
The latter two formulas which involve general coefficients
will be used below (see Section 4). Comparing (12) and
(13) yields
c1 = t+ t−1, c2 = −1.
With account of (19), this implies
b2 = 1, b1 = t
1
2 − t− 1
2 ,
which coincides with the coefficients in (4). Thus, our
statement is proved.
Since formulas (18) and (19) connect arbitrary pairs
b1, b2 and c1, c2, this allows us to gain a general skein
relation from the corresponding recurrence relation for
the set of torus knots T (s, 2).
3. Alexander Polynomials from Chebyshev
Polynomials
In this section, we describe the connection between
Alexander polynomials and Chebyshev polynomials, us-
ing the q-numbers. The q-number corresponding to the
integer n is defined as (see, e.g., [18, 19] and [20])
[n]q =
qn − q−n
q − q−1 , (20)
where q is a parameter. If q → 1, then [n]q → n. Consid-
ering q to be variable, we go over to the q-polynomials.
Some of the q-numbers (or q-polynomials) are as follows:
[1]q = 1, [2]q = q + q−1,
[3]q = q2 + 1 + q−2, [4]q = q3 + q + q−1 + q−3,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
[n]q=qn−1 + qn−3 + · · ·+ q−(n−1) =
n−1∑
i=0
qn−1−2i. (21)
The (easily verifiable) recurrence relation for the q-
numbers (or q-polynomials) is
[n+ 1]q = (q + q−1)[n]q − [n− 1]q. (22)
From now on, we rename the variable in the Alexander
polynomials:
t→ q.
For the considered class of torus knots, the compari-
son of the Alexander polynomials and the q-polynomials
implies that the Alexander polynomials (6) can be ex-
pressed through q-polynomials (21) in the simple way:
A2
n(q) = [n+ 1]q − [n]q. (23)
This relation for the Alexander polynomials was found
in [14, 15] in the context of their correspondence to the
masses of vector quarkonia.
Below, we will need some properties of the classi-
cal Chebyshev polynomials in order to formulate the
Alexander polynomials in terms of the Chebyshev ones.
If x = 2 cos θ, the Chebyshev polynomials of the first
kind are defined as
Tn(x) = 2 cos(nθ). (24)
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 131
A.M. GAVRILIK, A.M. PAVLYUK
From (24), some first cases read
T0=2, T1=x, T2=x2 − 2, T3=x3 − 3x, . . . .
The recurrence formula is known as
Tn+1 = xTn − Tn−1. (25)
Chebyshev polynomials of the second kind are
Vn(x) =
sin((n+ 1)θ)
sin θ
. (26)
Polynomials Tn and Vn are both monic and have the
degree n. From (26), we have
V0=1, V1=x, V2 = x2 − 1, V3 = x3 − 2x, . . . ,
and the recurrence relation is
Vn+1 = xVn − Vn−1. (27)
There is a connection between (24) and (26):
Tn(x) = Vn(x)− Vn−2(x). (28)
Putting
q = eiθ (29)
into (20), we have
[n]q =
sin(nθ)
sin θ
= Vn−1(x), (30)
where Vn(x) is the Chebyshev polynomial of the second
kind, and
x = 2 cos θ = q + q−1. (31)
From (30) and (31), it is seen that
Vn(q) = [n+ 1]q. (32)
Therefore, (23) takes the form
A2
n(q) = Vn(x)− Vn−1(x), x = q + q−1. (33)
Thus, the Alexander polynomials A2
n(q) are obtained
from the Chebyshev polynomials of the second kind
Vn(x) (26) after changing the variables x → q + q−1
by means of formula (33).
4. Generalized Alexander Polynomials and
HOMFLY Polynomials
Now let us consider the q,p -numbers, a natural gener-
alization of q-numbers. With the help of q, p-numbers,
we will construct a generalization of the Alexander poly-
nomials – A2
n(q, p) which now depend on two variables
q, p. Afterwards, we intend to show that A2
n(q, p) turn
into the well-known HOMFLY polynomials by an appro-
priate change of variables.
The q, p-number corresponding to the integer number
n is defined as (see e.g. [21])
[n]q,p =
qn − pn
q − p
, (34)
where q, p are some complex parameters. If p = q−1,
then [n]q,p = [n]q. Some of the q, p-numbers are
[1]q,p = 1, [2]q,p = q + p,
[3]q,p=q2 + qp+ p2, [4]q,p=q3 + q2p+ qp2 + p3,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
[n]q,p = qn−1 + qn−2p+ qn−3p2 + · · ·+
+qpn−2 + pn−1=
n−1∑
i=0
qn−1−ipi = qn−1
n−1∑
i=0
q−ipi. (35)
Considering q and p as variables, we deal with q, p-
polynomials. Then, the recurrence relation for them is
[n+ 1]q,p = (q + p)[n]q,p − qp[n− 1]q,p. (36)
On the base of Eq. (32) and expression (34) or (35)
for the q, p-polynomials, we introduce a natural general-
ization of the Chebyshev polynomials of the second kind
which now depend on the two variables:
Vn(q, p) = [n+ 1]q,p. (37)
From (36) and (37), the recurrence relation does follow:
Vn+1(q, p) = (q + p)Vn(q, p)− qpVn−1(q, p). (38)
Now, in analogy with (33), we introduce the two-variable
generalized Alexander polynomial as a linear combina-
tion of polynomials (37). Due to this proposal, the fol-
lowing recurrence formula takes place:
A2
n+1(q, p) = (q + p)A2
n(q, p)− qpA2
n−1(q, p). (39)
132 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
ON CHEBYSHEV POLYNOMIALS AND TORUS KNOTS
This is a direct analog of (7) and reduces to it if q = t
and p = t−1. To continue the analogy, we take
A2
0(q, p) = 1, A2
1(q, p) = q − qp+ p. (40)
It is easy to see that (39) with (40) will be valid if
A2
n(q, p) = Vn(q, p)− qpVn−1(q, p) =
= [n+ 1]q,p − qp[n]q,p.
(41)
Setting
q = reiθ, p = q̄ = re−iθ (42)
into (34), we have
[n]q,p =
rn sin(nθ)
r sin θ
= rn−1Vn−1(x). (43)
If r = 1, Eq. (43) turns into (30). Taking (37) and (43)
into account, we obtain Vn(q, p) with a factorized form
of the dependence on the variables r, x :
Vn(r, x) = rnVn(x). (44)
Here, Vn(x) is the classical Chebyshev polynomial of the
second kind, with x as in (31). The corresponding two-
variable Chebyshev polynomials of the first kind arise as
well:
Tn(r, x) = 2rn cos(nθ).
In the variables r, x, see (42) and (31), the recurrence
relation (39) can be written as
A2
n+1(r, x) = rxA2
n(r, x)− r2A2
n−1(r, x). (45)
The first two polynomials (40) become
A2
0(r, x) = 1, A2
1(r, x) = rx− r2. (46)
From (41) and (44), we also have
A2
n(r, x) = rn(Vn(x)− rVn−1(x)). (47)
Now we make a key proposal: we apply the generalized
Alexander polynomials A2
n(r, x) given by (45) and (46)
for describing the torus knots T (s, 2). From (19) with
account of (42), we have
c1 = rx, c2 = −r2,
and then
b2 = r, b1 = (rx− 2r)
1
2 = r
1
2 (x− 2)
1
2 . (48)
Hence, as a generalization of (1), from (10), (11) and
(48), we obtain the skein relation for the generalized
Alexander polynomials:
A+(r, x) = r
1
2 (x− 2)
1
2AO(r, x) + rA−(r, x). (49)
Now let us explore the connection between the gener-
alized Alexander skein relation (49) and the HOMFLY
skein relation. By definition, the HOMFLY polynomials
H(a, z) satisfy the skein relation
a−1H+(a, z)− a1H−(a, z) = zHO(a, z),
or, in equivalent form,
H+(a, z) = azHO(a, z) + a2H−(a, z), (50)
with Hunknot = 1. As before, consider the torus knots
T (s, 2), where s is an odd integer. For these, the notation
for the corresponding HOMFLY polynomials is similar
to that for the Alexander ones, namely
H(s, 2)(a, z) ≡ H(2m+ 1, 2)(a, z) ≡ H2
m(a, z) ≡ H2
m.
The short list of the HOMFLY polynomials for torus
knots T (s, 2) ≡ T (2m+ 1, 2) is:
H1,2
0 (a, z) = 1,
H3,2
1 (a, z) = 2a2 + a2z2 − a4,
H5,2
2 (a, z) = 3a4 + 4a4z2 + a4z4 − 2a6 − a6z2,
H7,2
3 (a, z) = 4a6 + 10a6z2 + 6a6z4 + a6z6 − 3a8−
−4a8z2 − a8z4,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · .
(51)
The recurrence relation for (51) reads
H2
m+1(a, z) = a2(z2 + 2)H2
m(a, z)− a4H2
m−1(a, z). (52)
If we compare (52) and (45), we see that, through the
substitution
r = a2, x = z2 + 2, (53)
the HOMFLY polynomials and the generalized Alexan-
der polynomials coincide:
H2
n(a, z) = A2
n(r, x) = rn(Vn(x)− rVn−1(x)).
Then, the HOMFLY skein relation (50) in the variables
r, x, see (53), looks as
H+(r, x) = r
1
2 (x− 2)
1
2HO(r, x) + rH−(r, x), (54)
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 133
A.M. GAVRILIK, A.M. PAVLYUK
which coincides with (49). In addition,
A2
0(r, x) = H2
0 (r, x), A2
1(r, x) = H2
1 (r, x).
Thus, we have proved that the generalized Alexan-
der polynomials and their skein relation go over into the
HOMFLY ones by applying parametrization (53). On
the other hand, the HOMFLY skein relation and poly-
nomials turn into the generalized Alexander ones with
the help of the inverse substitution
a = r
1
2 , z = (x− 2)
1
2 .
5. Concluding Remarks
We have demonstrated that the connection of the Cheby-
shev polynomials with the Alexander polynomials can
be realized in a rather simple way if one uses, as an
auxiliary tool, the concept of q-numbers. On the other
hand, the existence of the q,p -numbers, which general-
ize the q-numbers, makes it possible to generalize the
Chebyshev polynomials to their two-variable modifica-
tion and, by exploiting the analogy with the previous
one-variable case, also to achieve a two-variable general-
ization of the Alexander polynomials. Finally, we have
found that the two-variable extended Alexander polyno-
mials are mapped onto the HOMFLY polynomials.
We hope that the proposed way to use the Chebyshev
polynomials will be helpful for the further investigation
of knots and links, not only on the base of the Alexander
polynomials (along with their two-variable modification)
and the HOMFLY polynomials treated above, but also
possibly in connection with Kauffman polynomials and
other known polynomial invariants. In addition, it is of
interest to study, within the proposed scheme, the more
general (s, r) torus knots than those in the particular
class (s, 2) considered in this paper. Subsequently, we
hope to use the explored polynomial invariants within
the framework of some physical models.
This research was partially supported by the Grant
29.1/028 of the State Foundation of Fundamental Re-
search of Ukraine and by the Special Program of the Di-
vision of Physics and Astronomy of the NAS of Ukraine.
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Received 22.10.09
ПРО ПОЛIНОМИ ЧЕБИШОВА I ТОРИЧНI ВУЗЛИ
О.М. Гаврилик, А.М. Павлюк
Р е з ю м е
У роботi показано, що q-числа та їх двопараметричнi уза-
гальнення, q, p-числа можна використати для отримання де-
яких полiномiальних iнварiантiв торичних вузлiв i зачеплень.
По-перше, показано, що q-числа, якi тiсно пов’язанi з полi-
номами Чебишова, можуть бути пов’язанi з полiномами Але-
ксандера для класу T (s, 2) торичних вузлiв, де s – непар-
не цiле число, i використанi для знаходження вiдповiдного
скейн-спiввiдношення. Потiм використано цю процедуру для
отримання за допомогою q, p-чисел, двопараметричних уза-
гальнених полiномiв Александера та показано зв’язок остан-
нiх iз полiномiальними iнварiантами HOMFLY та їх скейн-
спiввiдношенням.
134 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
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