282
P. Ramadevi and Zodinmawia
3 Twist Knots
The colored Jones polynomial of twist knots K p with 2p half-twists [17] has a
multi-sum expression. We observed that the summand consists of the polynomial
of the trefoil K 1 = 3 1 for p > 0 (K −1 = 4 1 for p < 0 ) as the main body and a
twisting factor. With this structure in mind, the superpolynomials for trefoil 3 1 and
figure-eight 4 1 [14, 15] can be seen in a more succinct form:
P n (K 1 ; a, q, t) = (−t)
−n+1
∞
k=0
q
k (−atq −1 ; q) k
(q; q) k
q
1−n
; q
k
− at
3 q
n−1
; q
k
≡ (−t)
−n+1
∞
k=0
q
k H n,k ≡ (−t)
−n+1
∞
k=0
F n,k (a, q, t),
(16)
P n (K −1 ; a, q, t) =
∞
k=0
− at
2
−k q
−k(k−3)/2 H n,k ≡
∞
k=0
G n,k (a, q, t),
(17)
where we use the q-Pochhammer symbol (z; q) k =
k−1
j =0 (1 − zq j ). We expect
the superpolynomial for twist knots K p for |p| > 1 involving the above summand
multiplied by twisting factors:
P n (K p>0 ; a, q, t) = (−t)
−n+1
∞
s |p| ··· 1
F n,s p (a, q, t)×Twisting Factor, (18)
P n (K p<0 ; a, q, t) =
∞
s |p| ··· 1
G n,s p (a, q, t) × Twisting Factor.
(19)
The form of the twisting factor can be conjectured from the colored superpolynomials of 5 2 = K 2 and 6 1 = K −2 calculated in refs. [4, 18] up to n = 3. From this
data we could guess the form of the twisting factor for |p| = 2 and generalize to
arbitrary |p|. We have checked our conjectured superpolynomials (See eqns.(2.18,
2.19) in ref. [19]) with known results in the literature.
In order to study asymptotic expansion of superpolynomials, it is important to
reduce the multi-summation to minimum number of summation. We succeeded in
converting the multi-summation into a double summation using Bailey identities
(see appendix A in ref. [19]):
P n (K p>0 ; a, q, t) = (−t)
−n+1
∞
k=0
k
q
k
− atq −1 ; q
k
(q; q) k
(q
1−n
; q) k (−at
3 q
n−1
; q) k
× (−1)
at
2
pp q
(p+1/2))((−1) 1 − at 2 q 2
(at 2 q −1 ; q) k+1
k
q
.
(20)
P. Ramadevi and Zodinmawia
3 Twist Knots
The colored Jones polynomial of twist knots K p with 2p half-twists [17] has a
multi-sum expression. We observed that the summand consists of the polynomial
of the trefoil K 1 = 3 1 for p > 0 (K −1 = 4 1 for p < 0 ) as the main body and a
twisting factor. With this structure in mind, the superpolynomials for trefoil 3 1 and
figure-eight 4 1 [14, 15] can be seen in a more succinct form:
P n (K 1 ; a, q, t) = (−t)
−n+1
∞
k=0
q
k (−atq −1 ; q) k
(q; q) k
q
1−n
; q
k
− at
3 q
n−1
; q
k
≡ (−t)
−n+1
∞
k=0
q
k H n,k ≡ (−t)
−n+1
∞
k=0
F n,k (a, q, t),
(16)
P n (K −1 ; a, q, t) =
∞
k=0
− at
2
−k q
−k(k−3)/2 H n,k ≡
∞
k=0
G n,k (a, q, t),
(17)
where we use the q-Pochhammer symbol (z; q) k =
k−1
j =0 (1 − zq j ). We expect
the superpolynomial for twist knots K p for |p| > 1 involving the above summand
multiplied by twisting factors:
P n (K p>0 ; a, q, t) = (−t)
−n+1
∞
s |p| ··· 1
F n,s p (a, q, t)×Twisting Factor, (18)
P n (K p<0 ; a, q, t) =
∞
s |p| ··· 1
G n,s p (a, q, t) × Twisting Factor.
(19)
The form of the twisting factor can be conjectured from the colored superpolynomials of 5 2 = K 2 and 6 1 = K −2 calculated in refs. [4, 18] up to n = 3. From this
data we could guess the form of the twisting factor for |p| = 2 and generalize to
arbitrary |p|. We have checked our conjectured superpolynomials (See eqns.(2.18,
2.19) in ref. [19]) with known results in the literature.
In order to study asymptotic expansion of superpolynomials, it is important to
reduce the multi-summation to minimum number of summation. We succeeded in
converting the multi-summation into a double summation using Bailey identities
(see appendix A in ref. [19]):
P n (K p>0 ; a, q, t) = (−t)
−n+1
∞
k=0
k
q
k
− atq −1 ; q
k
(q; q) k
(q
1−n
; q) k (−at
3 q
n−1
; q) k
× (−1)
at
2
pp q
(p+1/2))((−1) 1 − at 2 q 2
(at 2 q −1 ; q) k+1
k
q
.
(20)