ON THE MODULARITY OF THE UNIFIED WRT INVARIANTS OF CERTAIN
SEIFERT MANIFOLDSKATHRIN BRINGMANN, KAZUHIRO HIKAMI AND JEREMY LOVEJOYFor Dennis Stanton on his 60th birthday
Abstract. We compute the unified WRT invariants of the Seifert manifolds M (2, 3, 8) and M (2, 3, 4)(arising from ±2 surgery on the trefoil knot). The first is essentially a mock theta function which isa piece of one of Ramanujan’s third order mock theta functions. The second is essentially the sumof a modular form and a false theta function.

1. Introduction and statement of results
Starting with the work of Lawrence and Zagier [16], many mock theta functions have been shownto coincide asymptotically with Witten-Reshetikhin-Turaev (WRT) invariants of Seifert manifolds(see [14]). Recently a family of unified WRT invariants, while technically not mock theta functions,were shown to have Hecke-type expansions closely resembling those of mock theta functions [15].
For example, related to the Poincaré homology sphere (c.f. [12, 17]) is the q-seriesX
M1 (q) :=q n (q n )n = 1 + q + q 3 + q 7 − q 8 − q 14 − q 20 − q 29 + q 31 + · · · ,(1.1)
n≥0which has the Hecke-type expansion [15, Eq. (3.36)]

XXX1

 (−1)r+s q r(3r+1)/2+s(3s+1)/2+2rs .
−q n (q n )n =(q)∞n≥0r,s≥0

(1.2)

r,s<0

Here we have employed the standard q-series notation(a)n := (a; q)n := (1 − a)(1 − aq) · · · (1 − aq n−1 ).
The right-hand side of (1.2) resembles expansions for mock theta functions (see [24]), but the quadratic form in the exponent of q is positive definite instead of indefinite. In fact, M1 (q) is a false
theta function.
A natural question is then whether any unified WRT invariants will turn out to be genuine mocktheta functions. Here we give one such example. We compute the unified WRT invariant of theSeifert manifold M (2, 3, 8) (arising from +2 surgery on the trefoil) and show that it is a mock thetafunction.
Date: July 16, 2012.
The first author was partially supported by NSF grant DMS-0757907.
The second author was partially supported by Grant-in-Aid for Scientific Research (21654053) from the Ministryof Education, Culture, Sports, Science and Technology of Japan.
The third author was partially supported by the French Agence Nationale de Recherche ANR-08-JCJC-0011.
12KATHRIN BRINGMANN, KAZUHIRO HIKAMI AND JEREMY LOVEJOY

Define φ0 (q) byX
q n (−q)2n+1 = 1 + 2q + 2q 2 + 3q 3 + 5q 4 + 6q 5 + 8q 6 + 11q 7 + 13q 8 + · · · ,φ0 (q) :=(1.3)

n≥0and recall Ramanujan’s third order mock theta function ψ(q),2ψ(q) :=Xn≥1qn= q + q 2 + q 3 + 2q 4 + 2q 5 + 2q 6 + 3q 7 + 3q 8 + · · · .
(q; q 2 )n(1.4)

Theorem 1.1. The unified WRT invariant of M (2, 3, 8) is√ 1/42q
φ (−q 1/2 ).
(1 − q) 0
Moreover, φ0 (q) is a mock theta function satisfying
2q 2 φ0 (q 2 ) = ψ(q) + ψ(−q).

(1.5)

We also consider the case of −2 surgery on the trefoil. Here we do not encounter a mock thetafunction, but the sum of a modular form and a false theta function. Define M (q) byX
M (q) :=
q 2n (−q)2n+1 = 1 + q + q 2 + q 3 + 2q 4 + 3q 5 + 3q 6 + 4q 7 + 5q 8 + · · · .
(1.6)
n≥0Theorem 1.2. The unified WRT invariant of M (2, 3, 4) is√ 1/42q
M (−q 1/2 ).
(1 − q)Moreover, M (q) satisfies,
2 + 2q 2 M (q) = (−q)∞ +X

q n(3n+1)/2 (1 − q 2n+1 ).

(1.7)

n≥0

2. Proof of Theorem 1.1
The WRT invariant τN (M ) for a 3-manifold M is constructed from the colored Jones polynomial
JN (K; q) for the knot K to be surgered [21]. To compute τN (M ) explicitly, useful is the cyclotomic
expansion of the colored Jones polynomial,JN (K; q) =∞X

CK (n)(q 1+N )n (q 1−N )n .

(2.1)

n=0

Here we normalize the colored Jones polynomial to be JN (unknot; q) = 1. We mean J1 (K; q) = 1,
and the N = 2 case corresponds to the Jones polynomial. Habiro proved that for arbitrary K wehave CK (n) ∈ Z[q, q −1 ] [13].
For example, we suppose that the 3-manifold M2 is constructed by +2-surgery on a knot K. Then
we have H1 (M2 ; Z) = Z2 , and the SU(2) WRT invariant τN (M2 ) for M2 is computed as [21]∞X

13/4(1 − ζN )τN (M2 ) = − √ (1 + (−1)N )ζN
CK (n) L2:N (q N )n+1 (q −N )n+1 q=ζ .
N
2 2n=0(2.2)

ON THE MODULARITY OF THE UNIFIED WRT INVARIANTS OF CERTAIN SEIFERT MANIFOLDS

Here ζN = e2πi/N , and L denotes the Laplace transform [4]. Explicitly we have
n+111
L2:N (q N )n+1 (q −N )n+1 = 2(−1)n+1 q − 2 (1 − q 2 )(−q; −q 2 )2n .

3(2.3)

Thus the unified WRT invariant Iq (M2 ) for M2 is written as(1 − q)Iq (M2 ) =√12q 4∞Xn

1

1

CK (n)(−1)n q − 2 (q 2 ; −q 2 )2n+1 .

(2.4)

n=0

We set M to be the Seifert manifold M (2, 3, 8) (see, e.g., [20]), which is obtained from +2-surgery
on the trefoil. Applying (2.4) to the colored Jones polynomial for the trefoil [11, 17]∞
XJN (trefoil; q) =q n (q 1+N )n (q 1−N )n ,(2.5)
n=0

we obtain(1 − q)Iq (M ) =√12q 4∞Xn1

1(−1)n q 2 (q 2 ; −q 2 )2n+1 .

(2.6)

n=0

This establishes the first part of Theorem 1.1.
Now to prove that φ0 (q) is a mock theta function, one is tempted to apply the results in [24],after showing thatn(−q)∞ X X
22φ0 (q) =(−1)j q 4n +7n−3j −5j (1 − q 2n+2 )(q)∞n≥0 j=−n−1(2.7)
XX(−q)∞ (r−s−1)/2 (r+s+1)2 /4+(3r+2)s−1−q= (−1)(q)∞r,s<0r,s≥0r6≡s(mod 2)

r6≡s(mod 2)

using standard Bailey pair techniques in [2]. However, it turns out that we may prove (1.5) directly.
We use a combinatorial argument. Since n2 = 1 + 3 + · · · + (2n − 1) and 1/(q; q 2 )n is the generating
function for partitions into odd parts less then 2n + 1, it is clear that ψ(q) is the generating function
for partitions into odd parts without gaps, i.e., where all odd parts < 2n − 1 occur if 2n − 1 occurs.
(This was first observed by N. Fine [8, p.57].) Now considerX
q 2n+2 (−q 2 ; q 2 )2n+1 .
q 2 φ0 (q 2 ) =n≥0

We interpret this graphically. The term (−q 2 ; q 2 )2n+1 contributes a partition λ into distinct evenparts of size at most 4n + 2. We represent λ as rows of 2’s of length at most 2n + 1. The term
q 2n+2 contributes a row of 2n + 2 ones, which we place above λ. For example, if n = 4 andλ = (16, 12, 8, 4, 2), we have1111111111
22222222
222222
2222
22
2

4

KATHRIN BRINGMANN, KAZUHIRO HIKAMI AND JEREMY LOVEJOY

Reading the columns, we obtain a partition into 2n + 2 odd parts without gaps, the fact that there
are no gaps coming from the fact that the rows of twos are of unequal length. In the example, weobtain (11, 9, 7, 7, 5, 5, 3, 3, 1, 1). Since the number of columns is even, the number being partitioned
is even. Thus q 2 φ0 (q 2 ) is the “even part” of ψ(q), i.e. (ψ(q) + ψ(−q))/2. This proves (1.5).
Now using the modern definition of a mock theta function as the holomorphic part of a harmonic
Maass form [5, 23, 24], it is clear that if f (q) is a mock theta function, then f (q) + f (−q) is either amock theta function or a modular form. To finish the proof of Theorem 1.1 we need to verify that
ψ(q) + ψ(−q) is not modular. (It is worth pointing out that it can happen that f (q) + f (−q) is
modular when f (q) is mock, for example when f is the second order mock theta function B(q) [10,Section 8].) To see this, recall that a mock theta function is modular iff its shadow (see [23] for thedefinition) is zero. Using [22, p. 65] and [24, Chapter 2], one may compute that ψ(q) has (up to anon-zero constant)
X(6n + 1)q n(3n+1)/2n∈Z

as shadow. From this one can conclude that ψ(q) + ψ(−q) has (up to a non-zero constant) theshadow
X(2.8)(6n + 1) 1 + (−1)n(3n+1)/2 q n(3n+1)/2 .
n∈Z

One can easily verify that (2.8) is nonzero.
Before continuing, we wish to make some remarks. First, there is a companion to φ0 (q) which is
the “odd part” of ψ(q). Define φ1 (q) byX
φ1 (q) =q n (−q)2n = 1 + q + 2q 2 + 3q 3 + 4q 4 + 5q 5 + 7q 6 + 9q 7 + 12q 8 + · · · .
(2.9)
n≥0

Arguing as above it is easy to see that
2qφ1 (q 2 ) = ψ(q) − ψ(−q)(2.10)and that ψ(q) − ψ(−q) is not modular, but mock.
Next, while we have opted for the combinatorial argument above, (1.5) and (2.10) can also be
deduced from the q-series identity
X bn q n(n+1)/2X= bq(−bq/t)n (tq)n ,(2.11)(tq)nn≥1

n≥0which follows from [8, Eq. (6.1)] or by counting partitions into distinct parts in two different ways.
Namely, if we let G(b, t; q) denote the left-hand side of (2.11), then we haveψ(q) + ψ(−q) = 2G(1/q, 1/q; q 2 ) + 2G(−1/q, −1/q; q 2 )X
X=2q n (−q 2 ; q 2 )n−1 + 2(−1)n q n (−q 2 ; q 2 )n−1 (by (2.11))n≥1=2Xn≥1q 2n (−q 2 ; q 2 )2n−1

n≥1= 2q 2 φ0 (q 2 ).

A similar argument yields (2.10).

ON THE MODULARITY OF THE UNIFIED WRT INVARIANTS OF CERTAIN SEIFERT MANIFOLDS

5If we allow more complicated series involving multiple sums and the q-binomial coefficients, (q)nn=k q(q)k (q)n−kthen this kind of argument can be applied to other mock theta functions. For a simple example,consider Ramanujan’s fifth order mock theta function
X q 2n2 +2n.
F1 (q) :=(q; q 2 )n+1n≥0

Define H(t; q) byX q 2n2 +2nH(t; q) :=,(tq; q 2 )n+1n≥0

so that F1 (q) ± F1 (−q) = H(1; q) ± H(−1; q). Then by [1, Eq. (3.37)] we haveXX
2n2 +2nm m n+mH(t; q) =qt q.
m q2n≥0m≥0Thus each ofX2q n +n+mm,n≥0andX

qn2 +n+m+1m,n≥0


n + 2m2m q(2.12)

n + 2m + 1
2m + 1 q(2.13)

is either modular or mock. We have verified that they are both indeed mock.
3. Proof of Theorem 1.2
In general when the manifold M−2 is obtained from −2 surgery on the knot K, we have∞X


1
1/4
CK (n) L−2:N (q N )n+1 (q −N )n+1 q=ζ ,(1 − ζN )τN (M−2 ) = √ (1 + (−1)N )ζNN2 2n=0

where the Laplace transformation (2.3) is replaced with [4]11
L−2:N (q N )n+1 (q −N )n+1 = 2(1 − q 2 )(−q; −q 2 )2n .

(3.1)

(3.2)

Thus the unified WRT invariant is given by(1 − q)Iq (M−2 ) =

√

2q

1
4∞
X

1

1

CK (n)(q 2 ; −q 2 )2n+1 .

(3.3)

n=0

In the case that K is the trefoil, whose colored Jones polynomial is given in (2.5), M−2 is the Seifert
manifold M (2, 3, 4), and substituting for CK (n) gives the first part of Theorem 1.2.
To confirm (1.7) we note that q 2 M (q) is the generating function for partitions into distinct parts
whose largest part is even. On the other hand, (−q)∞ is the generating function for all partitions
into distinct parts, and by Franklin’s involution (see [1]) we have that the final term in (1.7) is thegenerating function for partitions into distinct parts whose largest part is even minus the number

6

KATHRIN BRINGMANN, KAZUHIRO HIKAMI AND JEREMY LOVEJOYof partitions into distinct parts whose largest part is odd. This completes the proof of Theorem
1.2.
4. Concluding remarksIt is to be hoped that more unified WRT invariants will give rise to new mock theta functions, but
this remains to be seen. These invariants are not easy to compute and when they can be computed
they are often
only
 −1
 defined at roots of unity, since the coefficients CK (n) in (2.1) are only guaranteed
to be in Z q, q . The following describes one instance where (2.4) and (3.3) are convergent q-series.
The colored Jones polynomial for the twist knot Kp>0 is given by [18]p−1
∞YXsi (si +1) si+11+N
sp 1−Nq.
(4.1)) sp
JN (Kp ; q) =q (q)sp (qsi q
sp ≥···≥s2 ≥s1 ≥0

i=1

It is known [6] that ±2 surgery on the twist knot Kp gives a Seifert manifold. Applying the Laplacetransforms (2.3) and (3.2) to (4.1) we get the q-series related to the unified WRT invariants;

p−1∞
XY1
si+1
sp 12s(s+1)
ii
Ap (q) =q (q ; −q 2 )2sp +1,qsi q
sp ≥···≥s1 ≥0

Bp (q) =∞Xi=1

sp

11(−1)sp q 2 (q 2 ; −q 2 )2sp +1

sp ≥···≥s1 ≥0p−1
Yi=1

q si (si +1)


si+1.
si q

For now we have no futher information about these series for p ≥ 2 which would allow us to deduce
any automorphic properties.
Finally, if we would like combinatorial interpretations for the undilated φ0 (q) and φ1 (q) insteadof φ0 (q 2 ) and φ1 (q 2 ), then we may appeal to overpartitions [7]. Arguing as in Section 2, we find thatfor i = 0 or 1, the series qφi (q) is the generating function for overpartitions into odd parts such that(i) the non-overlined parts are without gaps and (ii) the overlined parts are less than 2k − 2i, wherek is the number of non-overlined parts. We point out that two eighth order/second order mock theta
functions of Gordon and McIntosh may also be interpreted in terms of overpartitions into odd parts
where the non-overlined parts are without gaps. These are [9, 19]
X q n2 (−q; q 2 )nV0 (q) := −1 + 2(q; q 2 )nn≥0

andV1 (q) :=X q (n+1)2 (−q; q 2 )nn≥0(q; q 2 )n+1.

All of this suggests that further series related to overpartitions where the non-overlined parts arewithout gaps are well worth investigating.
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[2] G.E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc. 293 (1986), 113-134.
[3] G.E. Andrews, F.J. Dyson, and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988),391–407.

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[14] K. Hikami, Mock (false) theta functions as quantum invariants, Regul. Chaotic Dyn. 10, No. 4 (2005), 509–530.
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127 (2003), 125–152.
[18] G. Masbaum, Skein-theoretical derivation of some formulas of Habiro, Algebr. Geom. Topol. 3 (2003), 537–556.
[19] R.J. McIntosh, Second order mock theta functions, Canad. Math. Bull. 50 (2007), 284-290.
[20] J. Milnor, On the 3-dimensional Brieskorn manifolds M (p, q, r), in “Knots, Groups, and 3-manifolds. Dedicatedto the Memory of R. H. Fox”, edited by L. P. Neuwirth, Princeton Univ. Press, pp.175–225 (1975).
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Math. 103 (1991), 547–597.
[22] G.N. Watson, The final problem: An account of the mock theta functions, J. London Math. Soc. 11 (1936), 55-80.
[23] D. Zagier, Ramanujan’s mock theta functions and their applications [d’aprés Zwegers and Bringmann-Ono],Séminaire Bourbaki, no 986.
[24] S. Zwegers, Mock Theta Functions, PhD Thesis, Universiteit Utrecht (2002).
Mathematisches Institut, Universität Köln, Weyertal 86-90, D-50931 Köln, Germany
E-mail address: kbringma@math.uni-koeln.de
Department of Mathematics, Naruto University of Education, Tokushima 772-8502, Japan
E-mail address: khikami@gmail.com
CNRS, LIAFA, Université Denis Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, FRANCE
E-mail address: lovejoy@liafa.jussieu.fr