284
P. Ramadevi and Zodinmawia
W(K p=2 ; z, w)
= − ln x ln(−t) −
π 2
3
+ iπ ln w + ln w(p(ln a + 2 ln t) + ln w
p+
1
2 )
+ Li 2 (x
−1 ) − Li 2 (x
−1 z) + Li 2 (−at) − Li 2 (−atz) + Li 2 (−at
3 x)
− Li 2 (−at
3 xz) − Li 2 (at
2 w) + Li 2 (at
2 wz) + Li 2 (w) + Li 2 (zw
−1 ).
(22)
The saddle point analysis of Eq. (21) in the limit ¯
h → 0 imposes the following
conditions:
∂
W(K p>0 ; z, w, x)
∂z
(z,w)=(z 0 ,w 0 )
= 0 =
∂
W(K p>0 ; z, w, x)
∂w
(z,w)=(z 0 ,w 0 )
. (23)
Further, the zero locus of the classical super-A-polynomial is determined by
y = exp
x
∂
W(K p>0 ; z 0 , w 0 , x)
∂x
.
(24)
Plugging the value of
W(K p=2 ; z, w)(22) into the above Eqs. (23) and (24), and
eliminating the extremum z 0 and w 0 , we obtained classical super-A-polynomials
(see Table 6 in ref. [19]).
In order to calculate the quantum super-A-polynomials, we have to find a
difference equation of minimal order for the colored superpolynomials. To find
the difference equations, we used the program iSumq.txt written by Xinyu
Sun, based on q-analogue of the algorithm developed in ref. [22] and obtained
ˆ
A super (5 2 ; ˆ
x, ˆ
y; a, q, t) (see Table 7 in ref. [19]). We checked that our results obey
ˆ
A super (5 2 ; ˆ
x, ˆ
y; a, q = 1, t) = A
super (5 2 ; x, y; a, t),
(25)
showing the validity of the categorified quantum volume conjecture for knot 5 2 .
Acknowledgments PR would like to acknowledge DST-RFBR grant(INT/RUS/RFBR/P-309)
for support. PR would like to thank Kavli Institute for Theoretical Physics at the University of
California Santa Barbara for local hospitality where initial parts of the manuscript were written.
PR would like to acknowledge the support in part by the National Science Foundation under Grant
No. PHY-1748958.
References
1. E. Witten, Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–
399 (1989). https://doi.org/10.1007/BF01217730
2. M. Khovanov, A categorification of the Jones polynomial. Duke Math. J. 101(02), 359–426
(2000). https://doi.org/10.1215/S0012-7094-00-10131-7
P. Ramadevi and Zodinmawia
W(K p=2 ; z, w)
= − ln x ln(−t) −
π 2
3
+ iπ ln w + ln w(p(ln a + 2 ln t) + ln w
p+
1
2 )
+ Li 2 (x
−1 ) − Li 2 (x
−1 z) + Li 2 (−at) − Li 2 (−atz) + Li 2 (−at
3 x)
− Li 2 (−at
3 xz) − Li 2 (at
2 w) + Li 2 (at
2 wz) + Li 2 (w) + Li 2 (zw
−1 ).
(22)
The saddle point analysis of Eq. (21) in the limit ¯
h → 0 imposes the following
conditions:
∂
W(K p>0 ; z, w, x)
∂z
(z,w)=(z 0 ,w 0 )
= 0 =
∂
W(K p>0 ; z, w, x)
∂w
(z,w)=(z 0 ,w 0 )
. (23)
Further, the zero locus of the classical super-A-polynomial is determined by
y = exp
x
∂
W(K p>0 ; z 0 , w 0 , x)
∂x
.
(24)
Plugging the value of
W(K p=2 ; z, w)(22) into the above Eqs. (23) and (24), and
eliminating the extremum z 0 and w 0 , we obtained classical super-A-polynomials
(see Table 6 in ref. [19]).
In order to calculate the quantum super-A-polynomials, we have to find a
difference equation of minimal order for the colored superpolynomials. To find
the difference equations, we used the program iSumq.txt written by Xinyu
Sun, based on q-analogue of the algorithm developed in ref. [22] and obtained
ˆ
A super (5 2 ; ˆ
x, ˆ
y; a, q, t) (see Table 7 in ref. [19]). We checked that our results obey
ˆ
A super (5 2 ; ˆ
x, ˆ
y; a, q = 1, t) = A
super (5 2 ; x, y; a, t),
(25)
showing the validity of the categorified quantum volume conjecture for knot 5 2 .
Acknowledgments PR would like to acknowledge DST-RFBR grant(INT/RUS/RFBR/P-309)
for support. PR would like to thank Kavli Institute for Theoretical Physics at the University of
California Santa Barbara for local hospitality where initial parts of the manuscript were written.
PR would like to acknowledge the support in part by the National Science Foundation under Grant
No. PHY-1748958.
References
1. E. Witten, Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–
399 (1989). https://doi.org/10.1007/BF01217730
2. M. Khovanov, A categorification of the Jones polynomial. Duke Math. J. 101(02), 359–426
(2000). https://doi.org/10.1215/S0012-7094-00-10131-7