On detection probabilities of link invariants
We prove that, for many standard link invariants, both the proportion of distinct invariant values and the detection probability among prime alternating links with at most n crossings decay exponentially in n, with an explicit universal rate. In fact, almost every such link belongs to an invariant fiber whose size is itself exponential in n. This phenomenon applies broadly, in particular to the Jones and HOMFLYPT polynomials and integral Khovanov homology. The companion website gives a much more detailed view of the data, including complete distributions of fiber sizes, separate alternating and non-alternating data, and topological data analysis.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Split probabilities and species tree inference under the multispecies coalescent model
Using topological summaries of gene trees as a basis for species tree inference is a promising approach to obtain acceptable speed on genomic-scale datasets, and to avoid some undesirable modeling assumptions. Here we st…
Fast and Efficient Calculations of Structural Invariants of Chirality
Chirality plays an important role in physics, chemistry, biology, and other fields. It describes an essential symmetry in structure. However, chirality invariants are usually complicated in expression or difficult to eva…
Symmetry DetectionStructured Abductive-Deductive-Inductive Reasoning for LLMs via Algebraic Invariants
Large language models exhibit systematic limitations in structured logical reasoning: they conflate hypothesis generation with verification, cannot distinguish conjecture from validated knowledge, and allow weak reasonin…
Logical ReasoningLearning Non-Linear Invariants for Unsupervised Out-of-Distribution Detection
The inability of deep learning models to handle data drawn from unseen distributions has sparked much interest in unsupervised out-of-distribution (U-OOD) detection, as it is crucial for reliable deep learning models. De…
Out-of-Distribution DetectionShape and Positional Geometry of Multi-Object Configurations
In previous work, we introduced a method for modeling a configuration of objects in 2D and 3D images using a mathematical "medial/skeletal linking structure." In this paper, we show how these structures allow us to captu…
Object