Twist Knot Invariants and Volume
Conjecture
P. Ramadevi and Zodinmawia
Abstract Chern–Simons theory provides a natural framework to construct a variety
of knot invariants. The calculation of colored HOMFLY-PT polynomials of knots
using SU (N) Chern–Simons theory requires the knowledge of 6j -symbols for
the quantum group U q (sl N ) which are not known for arbitrary representation.
Interestingly, our conjectured formula for superpolynomials (categorification of
colored HOMFLY-PT polynomials) of twist knots led to deducing closed form
expression for these symbols for a class of multiplicity-free U q (sl N ) representation.
Using the twist knot superpolynomials, we compute the classical and quantum
super-A-polynomials and test the categorified version of the quantum volume
conjecture.
Keywords Chern–Simons theory · Knot invariants · Colored HOMFLY-PT
polynomials · Quantum 6j -symbols · Superpolynomials · Volume conjectures ·
Super-A-polynomials
1 Introduction
The pioneering work of Witten [1] showed that Chern–Simons theory on a threesphere S 3 naturally describes knots and their invariants. The Chern–Simons action
CS(S 3 ) based on SU (N) gauge group is
CS(S
3 ) = k
S 3
Tr
A ∧ dA +
2
3
A ∧ A ∧ A
,
(1)
P. Ramadevi ()
Department of Physics, Indian Institute of Technology Bombay, Mumbai, India
e-mail: ramadevi@phy.iitb.ac.in
Zodinmawia
Department of Physics, Mizoram University, Aizawl, India
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_26
275
Conjecture
P. Ramadevi and Zodinmawia
Abstract Chern–Simons theory provides a natural framework to construct a variety
of knot invariants. The calculation of colored HOMFLY-PT polynomials of knots
using SU (N) Chern–Simons theory requires the knowledge of 6j -symbols for
the quantum group U q (sl N ) which are not known for arbitrary representation.
Interestingly, our conjectured formula for superpolynomials (categorification of
colored HOMFLY-PT polynomials) of twist knots led to deducing closed form
expression for these symbols for a class of multiplicity-free U q (sl N ) representation.
Using the twist knot superpolynomials, we compute the classical and quantum
super-A-polynomials and test the categorified version of the quantum volume
conjecture.
Keywords Chern–Simons theory · Knot invariants · Colored HOMFLY-PT
polynomials · Quantum 6j -symbols · Superpolynomials · Volume conjectures ·
Super-A-polynomials
1 Introduction
The pioneering work of Witten [1] showed that Chern–Simons theory on a threesphere S 3 naturally describes knots and their invariants. The Chern–Simons action
CS(S 3 ) based on SU (N) gauge group is
CS(S
3 ) = k
S 3
Tr
A ∧ dA +
2
3
A ∧ A ∧ A
,
(1)
P. Ramadevi ()
Department of Physics, Indian Institute of Technology Bombay, Mumbai, India
e-mail: ramadevi@phy.iitb.ac.in
Zodinmawia
Department of Physics, Mizoram University, Aizawl, India
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_26
275