JHEP09(2018)128
Notation, framing, normalization.
Notation. Throughout the text, we use the following notation: the quantum dimension
of the representation R is given by the celebrated hook formula
H
unknot
R
= D
R
=
Y
(i,j)∈R
{Aq
i−j
}
{q
h
i,j
}
(1.10)
where h
i,j
is the hook length for the element (i, j) of the Young diagram R, A = q
N
and
{x} := x − x
−1
, so that the quantum number [n] =
{q
n
}
{q}
. In the main body of the paper,
we give examples of this formula in the case of two-line Young diagrams, (4.20) and in
the case of representations from the adjoint tower, (4.17). Since we consider only finite-
dimensional representations of SU(N), each of them is associated with a Young diagram.
Throughout the text, we do not differ between the representation R and the corresponding
Young diagram. We also use the notation µ
Y
for the eigenvalues of the R-matrix in the
channel R ⊗
¯
R = ⊕Y and λ
Q
, in the channel R ⊗ R = ⊕Q.
Normalization. We denote by calligraphic letters the unreduced knot/link invariants: the
HOMFLY-PT invariant is denoted through H. The reduced HOMFLY-PT invariant, i.e.
that divided by the product of quantum dimensions of the representations coloring the link
components is denoted through just H. Note that the normalization of the link invariants
in [44] is different: as soon as they consider uncolored links, they normalize the invariants
dividing them just by one quantum dimension of the fundamental representation D
=
{A}
{q}
.
We also often need a mirror image of the knot/link, which, in terms of the invariant,
is described by the transformation (A, q) → (1/A, 1/q). We denote the corresponding
invariants with the bar:
H(A, q) := H
1
A
,
1
q
. (1.11)
Framing. Typically, all the invariants are given in the topological framing for knots and
in the standard (or canonical) framing for links [45–48]. This framing contains a trivial
U(1)-factor
U = q
2
P
i>j
|R
i
||R
j
|L
ij
/N
(1.12)
where the sum goes over all components of the link, the i-th component being colored
with the representation R
i
, and L
ij
is the linking number of the i-th and j-th components.
This factor is absent for knots, and we omit it for links unless the inverse is stated.
1
It is
essential only when one reduces representations, for the concrete SU(N), removing from
the Young diagrams the rows of the length N.
We sometimes need to consider various relations between knot/link invariants inspired
by group theory identities. Hence, these relations are most naturally realized in the framing
consistent with the group theory structure of the invariants, and, in the case of knots, this
is nothing but the vertical framing. In the case of links, this is the so called differential
framing, which is also consistent with the differential expansion of invariants, hence the
name [48]. We denote the invariants in this framing with a check-superscript:
∨
H.
1
It is often omitted, see [49, 50].
– 6 –
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