276
P. Ramadevi and Zodinmawia
where k is the coupling constant and A (gauge field) are the su(n) Lie algebra-valued
one forms: A =
a,μ A a
μ t a dx μ with t a denoting the generators of the gauge group.
For a knot K in S 3 , the corresponding Wilson loop operator (W LO) is obtained by
taking the holonomy of the gauge fields along the knot. More precisely, if T a
R are
the generators in a representation R ∈ SU (N), then the operator colored by R is
given by
W R [K] = Tr R exp
K
A
.
The expectation value of W LO, R [K]], are the knot invariants. In fact, the
R= [K]] is proportional to the Jones polynomial, J (K; q), for gauge group
SU (2) and HOMFLY-PT polynomial, P (K; a, q), for SU (N) gauge group. The
polynomial variables are related to coupling constant and rank of the group as
follows: a = q N and q = exp{[2πi/(k + N)]} [1]. For higher dimensional
representation R, R [K]] define the colored Jones J R (K; q) (R ∈ SU (2)) and
the colored HOMFLY-PT P R (K; a, q) (R ∈ SU (N)) polynomials. Collectively,
the knot invariants obtained from Chern–Simons theory are referred to as quantum
invariants of knots. In this note we focus on the colored HOMFLY-PT polynomials.
An intriguing property about quantum invariants of knots is the integrality
structure. We observe that the Jones polynomial for any knot has a Laurent series
expansion J (K; q) =
i c i q i , where {c i }’s are integers. Other quantum invariants
are also polynomials with integer coefficients. The quest to give a topological
answer to such an integrality property of Jones polynomials led Khovanov to
discover knot homology [2]. Particularly, Khovanov constructed a bigraded chain
complex (naturally associated with a planar diagram of a knot K) whose homology H sl 2
i,j (K) is invariant under the Reidemeister moves. Hence H sl 2
i,j (K) is an
invariant of the knot. More importantly, the Euler characteristics of this bigraded
homology are the Jones polynomial: J (K; q) =
i,j (−1) i q j dimH sl 2
i,j (K). Clearly,
dimH sl 2
i,j (K) must necessarily be an integer which provides a topological meaning
to the integers appearing in the Jones polynomial. Furthermore, a new, two-variable
invariant polynomial called Khovanov polynomial, Kh(K; q, t), can be constructed
by taking the graded Poincaré polynomial,
Kh(K; q, t) =
i,j
t
i q
j dimH sl 2
i,j (K).
(2)
Note that Jones polynomial is the t = −1 limit of Khovanov polynomial. In this
sense, Khovanov polynomial is the lift, or categorification, of the Jones polynomial.
The categorification of Jones polynomial by Khovanov led to the study of
homology theory for other quantum invariants. For the case of colored HOMFLY-PT
polynomial, a triply graded colored HOMFLY homology, (H HOMFLY
R
(K)) i,j,k , was
proposed in refs. [3–7] such that the graded Euler characteristic gives the colored
HOMFLY polynomial:
P. Ramadevi and Zodinmawia
where k is the coupling constant and A (gauge field) are the su(n) Lie algebra-valued
one forms: A =
a,μ A a
μ t a dx μ with t a denoting the generators of the gauge group.
For a knot K in S 3 , the corresponding Wilson loop operator (W LO) is obtained by
taking the holonomy of the gauge fields along the knot. More precisely, if T a
R are
the generators in a representation R ∈ SU (N), then the operator colored by R is
given by
W R [K] = Tr R exp
K
A
.
The expectation value of W LO, R [K]], are the knot invariants. In fact, the
R= [K]] is proportional to the Jones polynomial, J (K; q), for gauge group
SU (2) and HOMFLY-PT polynomial, P (K; a, q), for SU (N) gauge group. The
polynomial variables are related to coupling constant and rank of the group as
follows: a = q N and q = exp{[2πi/(k + N)]} [1]. For higher dimensional
representation R, R [K]] define the colored Jones J R (K; q) (R ∈ SU (2)) and
the colored HOMFLY-PT P R (K; a, q) (R ∈ SU (N)) polynomials. Collectively,
the knot invariants obtained from Chern–Simons theory are referred to as quantum
invariants of knots. In this note we focus on the colored HOMFLY-PT polynomials.
An intriguing property about quantum invariants of knots is the integrality
structure. We observe that the Jones polynomial for any knot has a Laurent series
expansion J (K; q) =
i c i q i , where {c i }’s are integers. Other quantum invariants
are also polynomials with integer coefficients. The quest to give a topological
answer to such an integrality property of Jones polynomials led Khovanov to
discover knot homology [2]. Particularly, Khovanov constructed a bigraded chain
complex (naturally associated with a planar diagram of a knot K) whose homology H sl 2
i,j (K) is invariant under the Reidemeister moves. Hence H sl 2
i,j (K) is an
invariant of the knot. More importantly, the Euler characteristics of this bigraded
homology are the Jones polynomial: J (K; q) =
i,j (−1) i q j dimH sl 2
i,j (K). Clearly,
dimH sl 2
i,j (K) must necessarily be an integer which provides a topological meaning
to the integers appearing in the Jones polynomial. Furthermore, a new, two-variable
invariant polynomial called Khovanov polynomial, Kh(K; q, t), can be constructed
by taking the graded Poincaré polynomial,
Kh(K; q, t) =
i,j
t
i q
j dimH sl 2
i,j (K).
(2)
Note that Jones polynomial is the t = −1 limit of Khovanov polynomial. In this
sense, Khovanov polynomial is the lift, or categorification, of the Jones polynomial.
The categorification of Jones polynomial by Khovanov led to the study of
homology theory for other quantum invariants. For the case of colored HOMFLY-PT
polynomial, a triply graded colored HOMFLY homology, (H HOMFLY
R
(K)) i,j,k , was
proposed in refs. [3–7] such that the graded Euler characteristic gives the colored
HOMFLY polynomial: