Tootfinder

Quantum equations for knots
Thomas Fiedler, Butian Zhang
https://arxiv.org/abs/2509.08423 https://arxiv.org/pdf/2509.08423…

Quantum equations for knots
This paper contains linear systems of equations which can distinguish knots without knot invariants. Let $M_n$ be the topological moduli space of all n-component string links and such that a fixed projection into the plane is an immersion. If a string link is the product of some string link diagram $T$ and the parallel n-cable of a framed long knot diagram $D$, then there is a canonical arc $push$ in $M_n$, defined by pushing $T$ through the n-cable of $D$. In this paper we apply the combinat…

Architecture of planetary systems with and without outer giant planets I. Inner planet detections around HD 23079, HD 196067, and HD 86226
J. -B. Delisle, J. P. Faria, D. S\'egransan, E. Fontanet, W. Ceva, D. Barbato, S. G. Sousa, N. Unger, A. Leleu, F. Bouchy, M. Cretignier, R. F. D\'iaz, X. Dumusque, Y. G. C. Frensch, N. C. Hara, G. Laughlin, G. Lo Curto, C. Lovis, M. Marmier, M. Mayor, L. Mignon, C. Mordasini, F. Pepe, N. C. Santos, S. Udry

Architecture of planetary systems with and without outer giant planets I. Inner planet detections around HD 23079, HD 196067, and HD 86226
Understanding the link between outer giant planets (OGPs) and inner light planets (ILPs) is key to understanding planetary system formation and architecture. The correlation between these two populations of planets is debated both theoretically -- different formation models predict either a correlation or an anticorrelation -- and observationally. Several recent attempts to constrain this correlation have yielded contradictory results, due to small-number statistics and heterogeneous samples. W…

Colored knot Floer homology: structures and examples
Akram Alishahi, Eugene Gorsky, Beibei Liu
https://arxiv.org/abs/2508.21776 https://arxiv.org/pdf/2508.…

Colored knot Floer homology: structures and examples
Inspired by the $S^n$ colored version of Khovanov and Khovanov-Rozansky homology, we define a colored version of knot Floer homology by studying the colimit of a directed system of link Floer homology with infinite full twists. Specifically, our $n$-colored knot Floer homology of a knot $K$ is then defined as the colimit of the link Floer homology of $(n, mn)$-cables of $K$ by fixing $n$ and letting $m$ goes to infinity. We show that the colimit of the infinite full twists is a module over the …