## Research

My research so far has been on the conjectural topological invariants, $\hat{Z}_b(q)$ and $F_K(x,q)$, which are integer coefficient $q$-series with interesting modular properties and are believed to have categorifications.
I'm working to find a mathematical definition for these invariants.

### Papers and preprints

*Large color $R$-matrix for knot complements and strange identities*

27 pages (arXiv:2004.02087 [math.GT])
*3d-3d correspondence for mapping tori* (with S. Chun, S. Gukov, N. Sopenko)

53 pages (arXiv:1911.08456 [hep-th])
*Higher rank $\hat{Z}$ and $F_K$*

17 pages (arXiv:1909.13002 [math.GT])

#### Coming soon :

*Large-N geometry of quantum modularity and $\hat{Z}$-invariants* (with T. Ekholm, A. Gruen, S. Gukov, P. Kucharski, P. Sułkowski)
*Refined $\hat{Z}$ and $F_K$* (with A. Gruen)

## What are $\hat{Z}_b(q)$ and $F_K(x,q)$?

Inspired by earlier works by many people (e.g. [LZ,H] among many others), the existence of a $q$-series valued invariant $\hat{Z}$ for 3-manifolds was conjectured in [GPV,GPPV].
More recently, an analog of $\hat{Z}$ for 3-manifolds with a toral boundary (e.g. knot complements) was studied in [GM], and it was named $F_K$ for the knot complement.

One of the reasons we are interested in this new conjectural invariant is its integrality (i.e. coefficients are integers) which hints that it may have a categorification.
Indeed, from physics point of view [GPV,GPPV], these invariants should be the graded Euler characteristics of certain Hilbert spaces of BPS states.
A hope is that categorification of these conjectural invariants (i.e. a 3-manifold analogue of Khovanov homology) will deepen our understanding of 3- and (smooth) 4-manifolds.

#### References

[LZ] R. Lawrence, D. Zagier - Modular forms and quantum invariants of 3-manifolds (1999)

[H] K. Hikami - On the Quantum Invariant for the Brieskorn Homology Spheres (2004)

[GPV] S. Gukov, P. Putrov, C. Vafa - Fivebranes and 3-manifold homology (2016)

[GPPV] S. Gukov, D. Pei, P. Putrov, C. Vafa - BPS spectra and 3-manifold invariants (2017)

[GM] S. Gukov, C. Manolescu - A two-variable series for knot complements (2019)