arXiv subjects
Publications and source records attributed to Hongjian Yang.
We establish a grading bound for colored $\mathfrak{gl}_N$ link homology that can be read off from any link diagram. In the context of contact geometry, this gives a bound on the Thurston--Bennequin number of a Legendrian link from $\mathfrak{gl}_N$ link homology, in the spirit of Ng.
We prove that Khovanov homology is an invariant of links in unparametrized $\mathbb{RP}^3$'s, i.e., oriented $3$-manifolds diffeomorphic to $\mathbb{RP}^3$. Along the way, we establish the functoriality of Khovanov homology for link cobordisms in $I\times\mathbb{RP}^3$.
The flip symmetry on knot diagrams induces an involution on Khovanov homology. We prove that this involution is determined by its behavior on unlinks; in particular, it is the identity map when working over $\mathbb{F}_2$. This confirms a folklore conjecture on the triviality of the Viro flip map. As a corollary, we prove that the symmetries on the transvergent and intravergent diagrams of a strongly invertible knot induce the same involution on Khovanov homology. We also apply similar techniques to study the half sweep-around map.
First step to investigate the potential of transfer learning applied to the field of spatial statistics
We study the instanton Floer homology for links in $\mathbb{RP}^3$ and compute the second page of Kronheimer--Mrowka's spectral sequence. As an application, we show that Khovanov homology detects the unknot and the projective unknot in $\mathbb{RP}^3$.
We define a homological action on sutured instanton Floer homology. This action is well-defined up to scalars, and behaves well under connected sums and sutured manifold decompositions. As an application, we show that instanton knot homology detects link splitting for two-component links.
Given an annular link $L$, there is a corresponding augmented link $\widetilde{L}$ in $S^3$ obtained by adding a meridian unknot component to $L$. In this paper, we construct a spectral sequence with the second page isomorphic to the annular Khovanov homology of $L$ and it converges to the reduced Khovanov homology of $\widetilde{L}$. As an application, we classify all the links with the minimal rank of annular Khovanov homology. We also give a proof that annular Khovanov homology detects unlinks.
Nutrient-based meal recommendations have the potential to help individuals prevent or manage conditions such as diabetes and obesity. However, learning people's food preferences and making recommendations that simultaneously appeal to their palate and satisfy nutritional expectations are challenging. Existing approaches either only learn high-level preferences or require a prolonged learning period. We propose Yum-me, a personalized nutrient-based meal recommender system designed to meet individuals' nutritional expectations, dietary restrictions, and fine-grained food preferences. Yum-me enables a simple and accurate food preference profiling procedure via a visual quiz-based user interface, and projects the learned profile into the domain of nutritionally appropriate food options to find ones that will appeal to the user. We present the design and implementation of Yum-me, and further describe and evaluate two innovative contributions. The first contriution is an open source state-of-the-art food image analysis model, named FoodDist. We demonstrate FoodDist's superior performance through careful benchmarking and discuss its applicability across a wide array of dietary applications. The second contribution is a novel online learning framework that learns food preference from item-wise and pairwise image comparisons. We evaluate the framework in a field study of 227 anonymous users and demonstrate that it outperforms other baselines by a significant margin. We further conducted an end-to-end validation of the feasibility and effectiveness of Yum-me through a 60-person user study, in which Yum-me improves the recommendation acceptance rate by 42.63%.