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Ka Ho Wong

Publications and source records attributed to Ka Ho Wong.

At least 19 recordsLinked to original sources
arXiv2026

Computation of leading coefficients in asymptotics of relative quantum invariants

We propose Asymptotic Expansion Conjectures of the relative Reshetikhin-Turaev invariants, of the relative Turaev-Viro invariants and of the discrete Fourier transforms of the quantum $6j$-symbols Our conjectures refine the corresponding volume conjectures by identifying not only the exponential terms with volume, but also the leading coefficients in the asymptotics expansions in terms of adjoint twisted Reidemeister torsions, logarithmic holonomies, determinant of Gram matrices and edge lengths. For families of special cases for which the exponential terms are already known, we prove our conjecture by computing the leading coefficients in the asymptotic expansions. The significance of these expansions is that we do not specify the way that the sequence of the colorings converges to the limit. As a consequence, the terms in the expansion will have to depend on the index $r,$ but the dependence is in a way that the terms are purely geometric invariants of the metrics on the underlying manifold and only the metrics vary with $r.$ Also, the appearance of the logarithmic holonomy and edge length terms in the relative setting is a new phenomenon that has never been observed in the expansion of either the colored Jones polynomials of links or the Reshetikhin-Turaev invariants of closed $3$-manifolds.

math.GT↗
arXiv2026

The Andersen-Kashaev volume conjecture for FAMED geometric triangulations

We investigate the Andersen-Kashaev volume conjecture by introducing the notion of FAMED triangulations, a class of ideal triangulations of $3$-manifolds satisfying certain specific combinatorial properties. For any FAMED triangulation of a one-cusped hyperbolic $3$-manifold $M$ with trivial second homology, we prove the existence of the Jones function in the Teichmüller TQFT of $M$. For FAMED geometric triangulations of $M$, we establish an asymptotic expansion of the Jones function in terms of the Neumann-Zagier potential function and the 1-loop invariant of Dimofte-Garoufalidis. As a consequence, we prove the Andersen-Kashaev volume conjecture for $M$ and provide new insights for the AJ conjecture for the Teichmüller TQFT developed by Andersen-Malusa. We further discover a new phenomenon: for FAMED geometric triangulations, the partition function in Teichmüller TQFT decays exponentially with decrease rate the hyperbolic volume of a cone structure determined by the prescribed angle structure. This perspective provides a potential application to the Casson conjecture on angle structures. Expanding the previous result of Guéritaud, Piguet-Nakazawa and the first author and complementing a parallel result of Guilloux and both authors, we prove all the above generalizations of the Andersen-Kashaev volume conjecture for every hyperbolic twist knot and for the first 42,000 hyperbolic knots in $S^3$.

math.GT↗
arXiv2025

Asymptotics aspects of Teichmüller TQFT for generalized FAMED semi-geometric triangulations

We introduce a generalized FAMED property for ideal triangulations of hyperbolic knot complements in $\mathbb{S}^3$. Given a hyperbolic knot $K$ in $\mathbb{S}^3$ and a semi-geometric triangulation $X$ of $\mathbb{S}^3 \setminus K$ that is generalized FAMED with respect to the longitude. We prove that in the semi-classical limit $\hbar \to 0^+$, for any angle structure $α$, the partition function $\mathscr{Z}_\hbar(X,α)$ in Teichmüller TQFT decays exponentially with decrease rate the volume of $\mathbb{S}^3 \setminus K$ equipped with a hyperbolic cone structure determined by $α$, and that the 1-loop invariant of Dimofte-Garoufalidis emerges as the 1-loop term. With additional combinatorial conditions on the triangulations, we prove the existence of the Jones function and show that its decay rate is governed by the Neumann-Zagier potential function. In particular, the Andersen-Kashaev volume conjecture holds for every hyperbolic knot whose complement admits such kinds of triangulations.

math.GT↗
arXiv2025

FAMED by computer: proving the Andersen-Kashaev volume conjecture for 42,000 knots

The FAMED condition is a combinatorial property for ideal triangulations of $3$-manifolds, which was introduced in 2024 by the first and last authors in order to study the Andersen--Kashaev volume conjecture. They notably proved that this conjecture is true for all FAMED geometric triangulations of one-cusped hyperbolic $3$-manifolds with trivial second homology. In this paper, using a straightforward computer implementation in Regina and Snappy, we find FAMED geometric triangulations for more than 42.000 complements of knots in $S^3$, including all knots with $12$ crossings or fewer and all knots whose complement can be triangulated with $23$ tetrahedra or fewer. As a consequence, the Andersen-Kashaev conjecture is now proven to be true for as many new examples. Along the way, we find several new insights about the FAMED property, which have great value in the quest of a general proof of the Andersen-Kashaev volume conjecture for every knot complement.

math.GT↗
arXiv2025

Detecting Neurocognitive Disorders through Analyses of Topic Evolution and Cross-modal Consistency in Visual-Stimulated Narratives

Early detection of neurocognitive disorders (NCDs) is crucial for timely intervention and disease management. Given that language impairments manifest early in NCD progression, visual-stimulated narrative (VSN)-based analysis offers a promising avenue for NCD detection. Current VSN-based NCD detection methods primarily focus on linguistic microstructures (e.g., lexical diversity) that are closely tied to bottom-up, stimulus-driven cognitive processes. While these features illuminate basic language abilities, the higher-order linguistic macrostructures (e.g., topic development) that may reflect top-down, concept-driven cognitive abilities remain underexplored. These macrostructural patterns are crucial for NCD detection, yet challenging to quantify due to their abstract and complex nature. To bridge this gap, we propose two novel macrostructural approaches: (1) a Dynamic Topic Model (DTM) to track topic evolution over time, and (2) a Text-Image Temporal Alignment Network (TITAN) to measure cross-modal consistency between narrative and visual stimuli. Experimental results show the effectiveness of the proposed approaches in NCD detection, with TITAN achieving superior performance across three corpora: ADReSS (F1=0.8889), ADReSSo (F1=0.8504), and CU-MARVEL-RABBIT (F1=0.7238). Feature contribution analysis reveals that macrostructural features (e.g., topic variability, topic change rate, and topic consistency) constitute the most significant contributors to the model's decision pathways, outperforming the investigated microstructural features. These findings underscore the value of macrostructural analysis for understanding linguistic-cognitive interactions associated with NCDs.

eess.AS↗
arXiv2024

Geometry of fundamental shadow link complements and applications to the 1-loop conjecture

We construct a geometric ideal triangulation for every fundamental shadow link complement and solve the gluing equation explicitly in terms of the logarithmic holonomies of the meridians of the link for any generic character in the distinguished component of the $\mathrm{PSL}(2;\mathbb{C})$-character variety of the link complement. As immediate applications, we obtain a new formula for the volume of a hyperideal tetrahedron in terms of its dihedral angles, and a formula for the volume of hyperbolic 3-manifolds obtained by doing Dehn-fillings to some of the boundary components of fundamental shadow link complements. Moreover, by using these ideal triangulations, we verify the 1-loop conjecture proposed by Dimofte and Garoufalidis for every fundamental shadow link complement. By using the result of Kalelkar-Schleimer-Segerman \cite{KSS}, we also prove the topological invariance of the 1-loop invariant and show that the 1-loop invariant satisfies a surgery formula. As a result, we prove the 1-loop conjecture for manifolds obtained by doing sufficiently long Dehn-fillings on boundary components of any fundamental shadow link complement. This verifies the 1-loop conjecture for a large class of hyperbolic 3-manifolds.

math.GT↗
arXiv2024

Measuring the Debye Energy in Superconductors via two Electron Photoemission Spectroscopy

We demonstrate theoretically that double angle resolved photoemission spectroscopy (2eARPES) can directly probe the existence of Cooper pairs away from the Fermi surface, and can thus provide insight into the characteristic energy scale around the Fermi surface, the Debye energy, in which electrons are bound into Cooper pairs. To this end, we compute the photoelectron counting rate $P^{(2)}$ in two different types of unconventional superconductors, a $d_{x^2-y^2}$-wave superconductor, and a topological superconductor with a broken time-reversal symmetry. We show that $P^{(2)}$ provides insight into the relative strength of intra- and inter-band pairing in multi-band systems, as well as into the spin polarization of the bands.

cond-mat.supr-con↗
arXiv2024

Generalized Bonahon-Wong-Yang volume conjecture of quantum invariants of surface diffeomorphisms I: the figure eight knot complement

We propose a generalization of the Bonahon-Wong-Yang volume conjecture of quantum invariants of surface diffeomorphisms, by relating the asymptotics of the invariants with certain hyperbolic cone structure on the mapping torus determined by the choice of the invariant puncture weights. We prove the conjecture for the once-punctured torus bundle with the diffeomorphism given by the word $LR$.

math.GT↗
arXiv2024

Resonant Chains and the Convergent Migration of Planets in Protoplanetary Disks

An increasing number of compact planetary systems with multiple planets in a resonant chain have been detected. The resonant chain must be maintained by convergent migration of the planets due to planet-disk interactions if it is formed before the dispersal of the protoplanetary gas disk. For type I migration in an adiabatic disk, we show that an analytic criterion for convergent migration can be developed by requiring that any part of the resonant chain should be convergently migrating toward the remaining part. The criterion depends primarily on the logarithmic gradients $α$ and $β$ of the surface density and temperature profiles of the disk, respectively, and it is independent of the absolute values of the surface density and temperature. The analytic criterion is applied to the Kepler-60, Kepler-80, Kepler-223, TOI-178, and TRAPPIST-1 systems. Due to the variation of planetary masses within the resonant chains, we find that convergent migration typically requires rather extreme values of $(α, β)$ that have little or no overlap with common disk models. Finally, we show that there is an empirical relationship between the distance of the innermost planet from the central star and the stellar mass for the observed resonant chain systems, which supports the idea that the resonant chains are formed and maintained by stalling the migration of the innermost planet near the inner edge of the disk truncated by the magnetic fields of the protostar.

astro-ph.EP↗
arXiv2023

Competing Higher Order Topological Superconducting Phases in Triangular Lattice Magnet-Superconductor Hybrid Systems

We demonstrate that a plethora of higher order topological phases emerge in magnet-superconductor hybrid (MSH) systems through the interplay of a stacked magnetic structure and an underlying triangular surface lattice; the latter being of great current experimental interest. Such lattices offer the ability to create three main types of edge terminations -- called x-, y- and y'-edges -- of MSH islands that, in turn, give rise to a complex phase diagrams exhibiting various regions of HOTSC phases. We identify the single adatom chain, as well as a pair of adjacent adatom chains (called a double-chain), as the basic building blocks for the emergence of HOTSC phases. Of particular interest are those HOTSC phase which arise from a competition between the topology of single and double-chain blocks, which are absent for square latices.

cond-mat.supr-con↗
arXiv2023

Adjoint twisted Reidemeister torsion and Gram matrices

We compute the adjoint twisted Reidemeister torsion for closed oriented hyperbolic $3$-manifolds and for hyperbolic $3$-manifolds with toroidal boundary. In our formula, we consider the manifold as obtained by doing a Dehn-filling along suitable boundary components of a fundamental shadow link complement, and the formula is in terms of the logarithmic holonomy of the meridians of the boundary components. As an important special case, we also write down a formula of the adjoint twisted Reidemeister torsion for the double of a hyperbolic $3$-manifold with totally geodesic boundary in terms of the edge lengths of a geometric ideal triangulation of the manifold. These unexpected formulas were inspired by, and played an important role in, the study of the asymptotic expansion of quantum invariants\,\cite{WY}.

math.GT↗
arXiv2022

Relative Reshetikhin-Turaev invariants, hyperbolic cone metrics and discrete Fourier transforms I

We propose the Volume Conjecture for the relative Reshetikhin-Turaev invariants of a closed oriented $3$-manifold with a colored framed link inside it whose asymptotic behavior is related to the volume and the Chern-Simons invariant of the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring. We prove the conjecture in the case that the ambient $3$-manifold is obtained by doing an integral surgery along some components of a fundamental shadow link and the complement of the link in the ambient manifold is homeomorphic to the fundamental shadow link complement, for sufficiently small cone angles. Together with Costantino and Thurston's result that all compact oriented $3$-manifolds with toroidal or empty boundary can be obtained by doing an integral surgery along some components of a suitable fundamental shadow link, this provides a possible approach of solving Chen-Yang's Volume Conjecture for the Reshetikhin-Turaev invariants of closed oriented hyperbolic $3$-manifolds. We also introduce a family of topological operations (the change-of-pair operations) that connect all pairs of a closed oriented $3$-manifold and a framed link inside it that have homeomorphic complements, which correspond to doing the partial discrete Fourier transforms to the corresponding relative Reshetikhin-Turaev invariants. As an application, we find a Poisson Summation Formula for the discrete Fourier transforms.

math.GT↗
arXiv2022

Competing topological superconducting phases in FeSe$_{0.45}$Te$_{0.55}$

We demonstrate that recent angle-resolved photo-emission spectroscopy experiments provide strong evidence for the existence of two competing topological superconducting phases in FeSe$_{0.45}$Te$_{0.55}$. The coupling of their underlying microscopic mechanisms -- one based on a three-dimensional topological insulator, one based on two-dimensional superconductivity -- stabilizes topological superconductivity over a wide range of parameters, and gives rise to two disjoint topological regions in the phase diagram of FeSe$_{0.45}$Te$_{0.55}$. We show that the topological origin of these regions can be identified by considering the form of Majorana edge modes at domain walls.

cond-mat.supr-con↗
arXiv2022

Higher Order Topological Superconductivity in Magnet-Superconductor Hybrid Systems

Quantum engineering of topological superconductors and of the ensuing Majorana zero modes might hold the key for realizing a new paradigm for the implementation of topological quantum computing and topology-based devices. Magnet-superconductor hybrid (MSH) systems have proven to be experimentally versatile platforms for the creation of topological superconductivity by custom-designing the complex structure of their magnetic layer. Here, we demonstrate that higher order topological superconductivity (HOTSC) can be realized in two-dimensional MSH systems by using stacked magnetic structures. We show that the sensitivity of the HOTSC to the particular magnetic stacking opens an unprecedented ability to tune the system between trivial and topological phases using atomic manipulation techniques. We propose that the realization of HOTSC in MSH systems, and in particular the existence of the characteristic Majorana corner modes, allows for the implementation of a measurement-based protocols for topological quantum computing.

cond-mat.supr-con↗
arXiv2022

Two-electron photoemission spectroscopy in Topological Superconductors

We demonstrate that the photo-electron counting rate, $P^{(2)}$, measured in two electron coincidence spectroscopy (2$e$-ARPES) experiments, provides unprecedented insight into the nature of topological superconductivity. In particular, we show that the spin dependence of $P^{(2)}$ allows one to detect superconducting spin-triplet correlations that are induced in a topological superconductor even in the absence of an associated triplet superconducting order parameter. This ability to detect spin-triplet correlations allows one to distinguish between two recently proposed scenarios for the microscopic origin of topological superconductivity in FeSe$_{0.45}$Te$_{0.55}$. Finally, we show that $P^{(2)}$ exhibits a characteristic intensity maximum that can be employed to detect topological phase transitions.

cond-mat.supr-con↗
arXiv2022

On the asymptotic expansion for the relative Reshetikhin-Turaev invariants of fundamental shadow link pairs

We study the asymptotic expansion conjecture of the relative Reshetikhin-Turaev invariants proposed in \cite{WY4} for all pairs $(M,L)$ satisfying the property that $M\setminus L$ is homeomorphic to some fundamental shadow link complement. The hyperbolic cone structure of such $(M,L)$ can be described by using the logarithmic holonomies of the meridians of some fundamental shadow link. We show that when the logarithmic holonomies are sufficiently small and all cone angles are less than $π$, the asymptotic expansion conjecture of $(M,L)$ is true. Especially, we verify the asymptotic expansion conjecture of the relative Reshetikhin-Turaev invariants for all pairs $(M,L)$ satisfying the property that $M\setminus L$ is homeomorphic to some fundamental shadow link complement, with cone angles sufficiently small. Furthermore, we show that if $M$ is obtained by doing rational surgery on a fundamental shadow link complement with sufficiently large surgery coefficients, then the cone angles can be pushed to any value less than $π$.

math.GT↗
arXiv2022

TalkTive: A Conversational Agent Using Backchannels to Engage Older Adults in Neurocognitive Disorders Screening

Conversational agents (CAs) have the great potential in mitigating the clinicians' burden in screening for neurocognitive disorders among older adults. It is important, therefore, to develop CAs that can be engaging, to elicit conversational speech input from older adult participants for supporting assessment of cognitive abilities. As an initial step, this paper presents research in developing the backchanneling ability in CAs in the form of a verbal response to engage the speaker. We analyzed 246 conversations of cognitive assessments between older adults and human assessors, and derived the categories of reactive backchannels (e.g. "hmm") and proactive backchannels (e.g. "please keep going"). This is used in the development of TalkTive, a CA which can predict both timing and form of backchanneling during cognitive assessments. The study then invited 36 older adult participants to evaluate the backchanneling feature. Results show that proactive backchanneling is more appreciated by participants than reactive backchanneling.

cs.HC↗