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John Chae

Publications and source records attributed to John Chae.

10 recordsLinked to original sources
arXiv2025

Quantum invariants of 3-manifolds and links: a review

We review the recent developments of quantum invariants of 3-manifolds and links: $\hat{Z}$ and $F_L$. They are $q$-series invariants originated from mathematical physics. They exhibit rich features, for example, quantum modularity, infinite dimensional Verma module structures and knot-quiver correspondence. Furthermore, they have connections to other topological invariants. We also provide a review of an extension of the above series invariants to Lie superalgebras.

math-ph↗
arXiv2025

A supergroup series for knot complements

We introduce a three variable series invariant $F_K (y,z,q)$ for plumbed knot complements associated with a Lie superalgebra $sl(2|1)$. The invariant is a generalization of the $sl(2|1)$-series invariant $\hat{Z}(q)$ for closed 3-manifolds introduced by Ferrari and Putrov and an extension of the two variable series invariant defined by Gukov and Manolescu (GM) to the Lie superalgebra. We derive a surgery formula relating $F_K (y,z,q)$ to $\hat{Z}(q)$ invariant. We find appropriate expansion chambers for certain infinite families of torus knots and compute explicit examples. Furthermore, we provide evidence for a non semisimple $Spin^c$ decorated TQFT from the three variable series. We observe that the super $F_K (y,z,q)$ itself and its results exhibit distinctive features compared to the GM series.

math.GT↗
arXiv2025

Link in $\mathbb{R}\mathbb{P}^3$ and the Topological Vertex

We provide the first computations of colored unknots and Hopf link in $\mathbb{R}\mathbb{P}^3$ using both the topological vertex and its refinement. Our approach utilizes the toric Calabi-Yau threefold arising from the geometric transition of the cotangent bundle of $\mathbb{R}\mathbb{P}^3$ under the large $N$ duality. We find that the link invariants are series in the Kahler parameters of the toric Calabi-Yau manifold and the $q$-expansions of the rational functions of the series have positivity property. We conjecture that they are Poincare series of an infinite dimensional link homology theory for links in $\mathbb{R}\mathbb{P}^3$. We compare our results with that of the $S^3$ and speculate the consequences of the series nature of the invariants.

math-ph↗
arXiv2023

A Cable Knot and BPS-Series II

This is a companion paper to earlier work of the author, which generalizes to an infinite family of $(2,2w+1)$-cabling of the figure eight knot ($|w|>3$) and proposes general formulas for the two-variable series invariant of the family of the cable knots. The formulas provide an insight into the cabling operation. We verify the conjecture through explicit examples using the recursion method, which also provide a strong evidence for the $q$-holonomic property of the series invariant. This result paves a road for computation of the WRT invariant of a 3-manifold obtained from Dehn surgery on the cable knots via a certain $q$-series. We also analyze and conjecture formulas for $(3,3w+1)$-cabling ($|w|>3$).

math.GT↗
arXiv2022

Fiber sum formulas for 4-manifolds, topological modular forms and $6d\ \mathcal{N}=(1,0)$ theories

Using the relation between four manifolds and topological modular form (TMF) from the six dimensional approach, we exhibit fiber sum formulas for infinite families of smooth spin four manifolds associated to compactifications of free and interacting 6d (1,0) SCFTs. We find that even the free theories have nontrivial fiber sum formulas and their forms are sensitive to an individual theory and parameters of four-manifolds. Furthermore, we reinforce the conjecture of Stolz and Teichner by expanding its evidence.

math-ph↗
arXiv2022

Witt invariants from q-series $\hat{Z}$

We present a relation between the Witt invariants of 3-manifolds and the $\hat{Z}$-invariants. It provides an alternative approach to compute the Witt invariants of 3-manifolds, which were originally defined geometrically in four dimensions. We analyze various homology spheres including a hyperbolic manifold using this method.

math.GT↗
arXiv2021

Towards a q-series for osp(2|2n)

A series invariant for a certain class of closed 3-manifolds associated with a type I Lie superalgebra sl(m|n) was introduced recently. We find a q-series for the other Lie superalgebra of the same type of the minimum rank.

math.GT↗
arXiv2021

A Cable Knot and BPS-Series

A series invariant of a complement of a knot was introduced recently. The invariant for several prime knots up to ten crossings have been explicitly computed. We present the first example of a satellite knot, namely, a cable of the figure eight knot, which has more than ten crossings. This cable knot result provides nontrivial evidence for the conjectures for the series invariant and demonstrates the robustness of integrality of the quantum invariant under the cabling operation. Furthermore, we observe a relation between the series invariant of the cable knot and the series invariant of the figure eight knot. This relation provides an alternative simple method for finding the former series invariant.

math.GT↗
arXiv2020

Knot Complement, ADO-Invariants and their Deformations for Torus Knots

A relation between the two-variable series knot invariant and the Akutus-Deguchi-Ohtsuki(ADO)-invariant was conjectured recently. We reinforce the conjecture by presenting explicit formulas and/or an algorithm for certain ADO-invariants of torus knots obtained from the series invariant of complement of a knot. Furthermore, one parameter deformation of ADO_3-polynomial of torus knots is provided.

math.GT↗