My research interests are in low (i.e. 3 and 4) dimensional topology, the Jones polynomial, hyperbolic geometry, mathematical physics, Chern-Simons theory, string theory, M-theory, enumerative combinatorics, enumerative algebraic geometry, number theory, quantum topology, asymptotic analysis, numerical analysis, integrable systems, motivic cohomology, K-theory, Galois theory, deformation and geometric quantization.
In my early career, I got interested in TQFT (topological quantum field theory) invariants of knotted 3-dimensional objects, such as knots, braids, srting-links or 3-manifolds.
Later on, I became interested in finite type invariants (a code name for perturbative quantum field theory invariants of knotted objects). I studied their axiomatic properties, and related the various definitions to each other. A side project was to study the various filtrations of the mapping class groups, and to explicitly construct cocycles, using finite type invariants.
More recently, I have been studying the colored Jones polynomials of a knot, and its limiting geometry and topology. The colored Jones polynomials is not a single polynomial, but a sequence of them, which is known to satisfy a linear q-difference equation. Writing the equation into an operator form, and setting q=1, conjecturally recovers the A-polynomial. The latter parametrizes out the moduli space of SL(2,C) representation of the knot complement.
Another relation between the colored Jones polynomial and SL(2,C) (ie, hyperbolic) geometry is the Volume Conjecture that relates evaluations of the colored Jones polynomial to the volume of a knot. This and related conjectures fall into the problem of proving the existence of asymptotic expansions of combinatorial invariants of knotted objects. Most recently, I am working on resurgence of formal power series of knotted objects. Resuregence is a key property which (together the nonvanishing of some Stokes constant) implies the Volume Conjecture. Resurgence is intimately related to Chern-Simons perturbation theory, and produces singularities of geometric as well as arithmetic interst. Resurgence seems to be related to the Grothendieck-Teichmuller group.
In short, my interests are in low dimensional topology, geometry and mathematical physics.
With J. Andersen and R. Kashaev, The volume conjecture for the KLV state-integral, preprint 2020.
With D. Zagier, Knots, perturbative series and quantum modularity, preprint 2020.
With J. Gu and M. Marino, The resurgent structure of quantum knot invariants, preprint 2020.
With N. Dunfield and H. Rubinstein, Counting essential surfaces in 3-manifolds, preprint 2020.
With J. Brown, T. Dimofte and N. Geer, The ADO invariants are a q-holonomic family, preprint 2020.
With R. van der Veen, The FKB invariant is the 3d index, preprint 2020.
With R. Kashaev, Resurgence of Faddeev's quantum dilogarithm, preprint 2020.
With R. Detcherry, A diagrammatic approach to the AJ Conjecture, Mathematische Annalen, 378 (2020), 447-484.
With D. Zagier, Asymptotics of Nahm sums at roots of unity, Ramanujan J., (2020) pp.
With C. Lee and R. van der Veen, The Slope Conjecture for Montesinos knots, International J. Mathematics, 31 (2020) no. 7, 2050056 66 pp.
With E. Getzler, Graph complexes and the symplectic character of the Torelli group, preprint 2017. Computer files: Symplectic character data
With R. Kashaev, A meromorphic extension of the 3D Index, Research in mathematical Sciences, 6 (2019) no. 1, 6:8.
With F. Calegari and D. Zagier, Bloch groups, algebraic K-theory, units and Nahm's Conjecture, preprint 2017.
With Z. Dansco, A construction of the graphic matroid from the lattice of integer flows, preprint 2016.
With I. Moffatt and Dylan P. Thurston, Non-peripheral ideal decompositions of alternating knots, preprint 2016.
With A. Lauda and T.T.Q. Le, The colored HOMFLY-PT polynomial is q-holonomic, Duke Math. J., 167 (2018) 397-447.
with C. Hodgson, N. Hoffman and H. Rubinstein, The 3D-index and normal surfaces, Illinois J. Mathematics, 60 (2016) 289-353.
With T.T.Q. Le, A survey of q-holonomic functions, L'Enseignement Mathematique, 62 (2016) 501-525.
With A. Reid, Constructing 1-cusped isospectral non-isometric hyperbolic 3-manifolds, J. Topology and Analysis, 10 (2018) 1-25. Computer files: Isospectral data
With P. Kucharski and P. Sulkowski, Knots, BPS states, and algebraic curves, Commun. Math. Physics, 346 (2016) 75-113.
With E. Sabo and S. Scott, Exact computation of the n-loop invariants of knots, Experimental Math. 25 (2016) 125-129. Computer files: n-loop data
With E. Fominykh, M. Goerner, V. Tarkaev and A. Vesnin, A census of tetrahedral hyperbolic manifolds, Experimental Math. 25 (2016) 466-481. Computer files: Tetrahedral census
With T. Dimofte, Quantum modularity and complex Chern-Simons theory, Commun. Number Theory Phys., 12 (2018), no. 1, 1â€“52.
With R. Kashaev, Evaluation of state integrals at rational points, Commun. Number Theory Phys., 9 (2015) 549-582.
With R. van der Veen, Quadratic integer programming and the slope conjecture, New York J. Mathematics, 22 (2016) 907-932.
The Newton polytope of a recurrent sequence of polynomials, preprint 2014.
Recurrent sequences of polynomials in 3-dimensional topology, Acta Mathematica Vietnamica, 39 (2014) 541-548.
With M. Goerner and C. Zickert, The Ptolemy field of 3-manifold-representations, Algebraic and Geometric Topology, 15 (2015) 371-397. Computer files: Unhyperbolic
With R. van der Veen, A generating series for Murakami-Ohtsuki-Yamada graph evaluations, Acta Mathematica Vietnamica, 39 (2014) 529-539.
With T. Vuong, A stability conjecture for the colored Jones polynomial, Topology Proceedings, 49 (2017) 211-249.
With C. Zickert, The symplectic properties of the PGL(n,C) gluing equations, Quantum Topology, 7 (2016) no. 3 505-551. Computer files: Unhyperbolic
With S. Norin and T. Vuong, Flag algebras and the stable coefficients of the Jones polynomial, European J. Combinatorics, 51 (2016) 165-189.
With R. Kashaev, From state integrals to q-series, Math. Res. Letters, 24 (2017) 781-801.
With T. Vuong, Alternating knots, planar graphs and q-series, Ramanujan J., 36 (2015) 501-527. Computer files: Phi_G data
The colored HOMFLY polynomial is q-holonomic, preprint 2012. Computer files: HOMFLY twist knot data
With C. Koutschan, Irreducibility of q-difference operators and the knot 7_4, Algebraic and Geometric Topology, 13 (2013) 3261-3286. Computer files: double twist data
with C. Hodgson, H. Rubinstein and H. Segerman, 1-efficient triangulations and the index of a cusped hyperbolic 3-manifold, Geometry and Topology, 19 (2015) 2619-2689. Computer files: 3D index data
The 3D index of an ideal triangulation and angle structures, Ramanujan J. 40 (2016) 573-604.
With M. Goerner and C. Zickert, Gluing equations for PGL(n,C)-representations of 3-manifolds, Algebraic and Geometric Topology, 15 (2015) 565-622. Computer files: Unhyperbolic
With T. Dimofte, The quantum content of the gluing equations, Geometry and Topology, 17 (2013) 1253-1315.
Quantum Knot Invariants, Arbeitstagung talk Bonn 2011, Research in the Mathematical Sciences, 5 (2018), no. 1, Paper No. 11, 17.
With D.P. Thurston and C. Zickert, The complex volume of SL(n,C) representations of 3-manifolds, Duke Math J., 164 (2015), no. 11, 2099-2160.
With C. Koutschan, Twisting q-holonomic sequences by complex roots of unity, ISSAC (2012) 179-186. Computer files: twisting qholonomic data
With T.T.Q. Le, Nahm sums, stability and the colored Jones polynomial, Research in Mathematical Sciences, 2 (2015) 1-55.
With N. Dunfield, Incompressibility criteria for spun-normal surfaces, Transactions of the AMS 364 (2012) 6109-6137.
With T. Mattman, The A-polynomial of the (-2,3,3+2n) pretzel knots, New York Journal of Mathematics 17 (2011) 269-279. Computer files: pretzel A-polynomial data
With C. Koutschan, The non-commutative A-polynomial of the (-2,3,n) pretzel knots, Experimental Mathematics 21:3 (2012) 241-251. Computer files: pretzel data
With C. Koutschan, The SL_3 Jones polynomial of the trefoil: a case study of q-holonomic sequences, Advances in Applied Mathematics 47 (2011) 829-839.
The Newton polygon of a recurrence sequence of polynomials and its role in TQFT, preprint 2010.
With H. Morton and T. Vuong, The SL_3 colored Jones polynomial of the trefoil, Proceedings of the AMS 141:6 (2013) 2209-2220.
With I. Popescu, Analyticity of the planar limit of a matrix model, Annales Henri Poincare, 14 (2013) 499-565.
What is a sequence of Nilsson type?, Contemporary Mathematics, 541 (2011) 145-157.
The degree of a q-holonomic sequence is a quadratic quasi-polynomial, Electronic Journal of Combinatorics 18:2 (2011) research article P4, 23 pages.
Knots and tropical curves, Contemporary Mathematics, 541 (2011) 83-111.
With R. van der Veen, Asymptotics of quantum spin networks at a fixed root of unity, Mathematische Annalen, 352 (2012) 987-1012.
With A. Its, A. Kapaev and M. Marino, Asymptotics of the instantons of Painleve I, Int. Math. Res. Not. IMRN, 3 (2012) 561-606.
The Jones slopes of a knot, Quantum Topology, 2 (2011) 43-69.
With J. Pommersheim, Sum-integral interpolators and the Euler-Maclaurin formula for polytopes, Transactions of the AMS, 364 (2012) 2933-2958.
With J. Dubois, Rationality of the SL(2,C)-Reidemeister torsion in dimension 3, Topology Proceedings, 47 (2016) 115-134.
With R. van der Veen, Asymptotics of classical spin networks, Geometry and Topology, 17 (2013) 1-37.
With M. Marino, Universality and asymptotics of graph counting problems in non-orientable surfaces, Journal of Combinatorial Theory A, 117 (2010) 715-740.
With X. Sun, A new algorithm for the recursion of hypergeometric multisums, Contemporary Mathematics, 517 (2011) 143-156. Computer files: Multisum data
With T.T.Q. Le and M. Marino, Analyticity of the free energy of a closed 3-manifold, SIGMA, 4 (2008) 080 20 pages.
With X. Sun, The non-commutative A-polynomial of twist knots, Journal of Knot Theory and Its Ramifications, 19 (2010) 1571-1595. Computer files: twist knot data
Chern-Simons theory, analytic continuation and arithmetic, Acta Mathematica Vietnamica, 33 (2008) 335-362.
G-functions and multisum versus holonomic sequences, Advances in Mathematics, 220 (2009) 1945-1955.
With J. Belissard, Algebraic generating functions of matrices over a group-ring, Chelyabinsk Math Journal, 3 (358) (2015) 50-61.
An ansatz for the asymptotics of hypergeometric multisums, Advances in Applied Mathematics, 41 (2008) 423-451.
An extended version of additive K-theory, Journal of K-Theory, 4 (2009) 391-403.
q-terms, singularities and the extended Bloch group, preprint 2007.
With O. Costin, Resurgence of the Euler-MacLaurin summation formula, Annales de l' Institut Fourier 58 (2008) 893-914.
With O. Costin, Resurgence of the fractional polylogarithms, Mathematical Research Letters, 16 (2009) no.5 817-826.
With O. Costin, Resurgence of the Kontsevich-Zagier power series, Annales de l' Institut Fourier 61 (2011) 1225-1258.
With T.T.Q. Le, Gevrey series in quantum topology, J. Reine Angew. Math., 618 (2008) 169-195.
With N. Dunfield, A. Shumakovitch and M. Thislethwaite, Behavior of knot invariants under genus 2 mutation, New York Journal of Mathematics 16 (2010) 99-123.
With T.T.Q. Le, Is the Jones polynomial of a knot really a polynomial? Proceedings of the KauffmanFest, Journal of Knot Theory and its Ramifications, 15 (2006) 1-18.
With M. Marino, On Chern-Simons Matrix Models, preprint 2006.
With T.T.Q. Le, Asymptotics of the colored Jones function of a knot, Geometry and Topology 15 (2011) 2135-2180.
With X. Sun, The C-polynomial of a knot, Algebraic and Geometric Topology, 6 (2006) 1001-1031.
With M. Loebl, A non-commutative formula for the colored Jones function, Mathematische Annalen, 336 (2006) 867-900.
With Y. Lan, Experimental evidence for the Volume Conjecture of the simplest hyperbolic non-2-bridge knot, Algebraic and Geometric Topology, 5 (2005) 379-403.
With N. Dunfield, The nontriviality of the A-polynomial for knots in S^3, Algebraic and Geometric Topology, 4 (2004) 1145-1153.
With J. Geronimo, Asymptotics of q-difference equations, Contemporary Math., 416 (2006) 83-114.
Does the Jones polynomial determine the signature of a knot? preprint 2003.
On the characteristic and deformation varieties of a knot, Proceedings of the Casson Fest, Geometry and Topology Monographs, 7 (2004) 291-309.
With T.T.Q. Le, The colored Jones function is q-holonomic, Geometry and Topology, 9 (2005) 1253-1293.
Difference and differential equations for the colored Jones function, Journal of Knot Theory and its Ramifications, 17 (2008) 495-510.
With T.T.Q. Le and D. Zeilberger, The quantum MacMahon Master Theorem, Proc. Natl. Academy of Sciences, 103 (2006) 13928-13931.
Links with trivial Alexander module and nontrivial Milnor invariants, Chelyabinsk Math Journal, 164 (2015) 209-2160.
With A. Kricker, A surgery view of boundary links, Math. Annalen, 327 (2003) 103-115.
With P. Teichner, On knots with trivial Alexander polynomial, Journal of Diff. Geometry, 67 (2004) 763-789.
With M. Loebl, Random walks and the colored Jones function, Combinatorica, 25 (2005) 651-671.
Periodicity of Goussarov-Vassiliev knot invariants, Geom. and Topology Monographs, 4 (2002) 43-54.
Beads: From Lie algebras to Lie groups, preprint 2002.
A conjecture on Khovanov's invariants, Fundamenta Mathematicae, 184 (2004) 99-101.
With A. Kricker, Finite type invariants of cyclic branched covers, Topology, 43 (2004) 1247-1283.
With A. Kricker, A rational noncommutative invariant of boundary links, Geometry and Topology, 8 (2004) 115-204.
With J. Levine, Concordance and 1-loop clovers, Algebraic and Geometric Topology, 1 (2001) 687-697.
With J. Levine, Analytic invariants of boundary links, Journal of Knot Theory and its Ramifications, 11 (2002) 283-293.
With J. Levine, Homology surgery and invariants of 3-manifolds, Geometry and Topology, 5 (2001) 551-578.
With M. Goussarov and M. Polyak, Calculus of clovers and finite type invariants of 3-manifolds, Geometry and Topology, 5 (2001) 75-108.
Whitehead doubling persists, Algebraic and Geometric Topology, 4 (2004) 935-942.
With L. Rozansky, The loop expansion of the Kontsevich integral, the null-move and S-equivalence, Topology, 43 (2004) 1183-1210.
The mystery of the brane relation, Journal of Knot Theory and its Ramifications, 11 (2002) 725-738.