Gukov-Manolescu series (F_K)
This table contains the the first few terms of the Gukov-Manolescu series under the column fk_polynomial. The braid and inversion datum used to compute the inverted state sum are listed under braid and inversion_data.
$$\text{WRT}_r(M) = \sum_c \prod_e d_c e^{2\pi i \cdot \text{CS}(c)/k} \qquad \widehat{Z}(M_3) = \sum_\alpha q^{h_\alpha} \cdot \psi_\alpha(q) \qquad J_N(K; q) = \text{Tr}_{V_N}(R_K) \qquad \Delta_K(t) = \sum_{i=0}^n a_i t^i \qquad \chi(Y) = \sum_i (-1)^i \dim H_i(Y) \qquad F_K = \partial \overline{X_K}$$
$$\langle M_3 \rangle = \int [DX] e^{iS_{CS}[A]} \qquad HF^\circ(Y) = \bigoplus_{\mathfrak{s}} HF^\circ(Y, \mathfrak{s}) \qquad \mathcal{Z}_{BPS}(M_3, q) = \sum_{\gamma} \Omega(\gamma) q^{\deg(\gamma)} \qquad P_K(v,z) = \langle K \rangle_{SL_2} \qquad V_K(t) = \sum_{n=0}^\infty V_n t^n \qquad \text{Kh}(L) = H^*(\text{CKh}(D))$$
$$\widehat{HF}(Y) = H_*(\widehat{CF}(Y)) \qquad \mathcal{L}(K) = \text{lk}(K, K') \qquad \tau(K) = \min\{s \in \mathbb{Z} \mid HFK_s(K) \neq 0\} \qquad \text{vol}(M) = \frac{1}{6\pi^2} \int_{M} \omega \qquad \text{CS}(A) = \frac{1}{4\pi} \int \text{Tr}(A \wedge dA) \qquad \mathfrak{sl}_n(q) = \sum_{k=0}^{n-1} q^k$$
$$S^3_K(\alpha) = S^3 \setminus \nu(K) \cup D^2 \times S^1 \qquad \mathcal{H}_*(Y) = \text{Tor}_{R[U]}(R, \widehat{CF}(Y)) \qquad \text{rk}(HF^+(Y)) = |H_1(Y)| \qquad \mathcal{A}(M) = \int_M \text{Tr}(R \wedge R) \qquad \nu(K) = \min(\text{crossing number}) \qquad \text{g}(K) = \frac{\deg(\Delta_K) + 1}{2}$$
$$\text{WRT}_r(M) = \sum_c \prod_e d_c e^{2\pi i \cdot \text{CS}(c)/k} \qquad \widehat{Z}(M_3) = \sum_\alpha q^{h_\alpha} \cdot \psi_\alpha(q) \qquad J_N(K; q) = \text{Tr}_{V_N}(R_K) \qquad \Delta_K(t) = \sum_{i=0}^n a_i t^i \qquad \chi(Y) = \sum_i (-1)^i \dim H_i(Y) \qquad F_K = \partial \overline{X_K}$$
$$\langle M_3 \rangle = \int [DX] e^{iS_{CS}[A]} \qquad HF^\circ(Y) = \bigoplus_{\mathfrak{s}} HF^\circ(Y, \mathfrak{s}) \qquad \mathcal{Z}_{BPS}(M_3, q) = \sum_{\gamma} \Omega(\gamma) q^{\deg(\gamma)} \qquad P_K(v,z) = \langle K \rangle_{SL_2} \qquad V_K(t) = \sum_{n=0}^\infty V_n t^n \qquad \text{Kh}(L) = H^*(\text{CKh}(D))$$
$$\widehat{HF}(Y) = H_*(\widehat{CF}(Y)) \qquad \mathcal{L}(K) = \text{lk}(K, K') \qquad \tau(K) = \min\{s \in \mathbb{Z} \mid HFK_s(K) \neq 0\} \qquad \text{vol}(M) = \frac{1}{6\pi^2} \int_{M} \omega \qquad \text{CS}(A) = \frac{1}{4\pi} \int \text{Tr}(A \wedge dA) \qquad \mathfrak{sl}_n(q) = \sum_{k=0}^{n-1} q^k$$
$$S^3_K(\alpha) = S^3 \setminus \nu(K) \cup D^2 \times S^1 \qquad \mathcal{H}_*(Y) = \text{Tor}_{R[U]}(R, \widehat{CF}(Y)) \qquad \text{rk}(HF^+(Y)) = |H_1(Y)| \qquad \mathcal{A}(M) = \int_M \text{Tr}(R \wedge R) \qquad \nu(K) = \min(\text{crossing number}) \qquad \text{g}(K) = \frac{\deg(\Delta_K) + 1}{2}$$
Caltech Quantum Topology Group
We study knots, 3-manifolds, and quantum invariants at the intersection of geometry, topology, and quantum field theory — with an emphasis on open computational data and reproducible research.
About our research
The Caltech Quantum Topology Group explores the rich interface between low-dimensional topology, quantum field theory, and representation theory. Our work focuses on understanding the deep connections between knot invariants, 3-manifold invariants, and their origins in physics — from Chern-Simons theory to categorification and beyond.
A central theme of our research is the use of computational experiments to discover new mathematical structures and test conjectures. We believe in open science: our datasets, code, and computational results are freely available to support both theoretical advances and emerging AI-driven approaches to mathematical discovery.
Research in the group has been supported in part by the Simons Collaboration on New Structures in Low-Dimensional Topology, the Simons Foundation, and the National Science Foundation.
Data & software
Open datasets and tools for working with knots, links, and 3-manifolds. Use them in your own experiments, or as benchmarks for new conjectures and models.
Normalized braids
Canonical braids up to 10 crossings, lexicographically minimal among cyclic rotations and mirror images, with each generator appearing at least twice.