arXiv research

Classifies doodles into prime and super prime types, describing them with doodle codes.

Alexander invariant created for doodles, vanishes on unlinked doodles.

Doodles were introduced in [R. Fenn and P. Taylor, Introducing doodles, Topology of low-dimensional manifolds, pp. 37--43, Lecture Notes in Math., 722, Springer, Berlin, 1979] but were restricted to embedded circles in the 2-sphere. Khovanov, [M. Khovanov, Doodle groups, Trans. Amer. Math. Soc. 349 (1997), 2297--2315],…

Doodles link to commutator identities in a 2-sphere.

Paper introduces skew-symmetric matrices for virtual doodle classification.

Complete invariant defined for doodles on a sphere.

We discuss Gauss codes of virtual diagrams and virtual doodles. The notion of a left canonical Gauss code is introduced and it is shown that oriented virtual doodles are uniquely presented by left canonical Gauss codes.

A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation generated by flat version of classical Reidemesiter moves and virtual Reidemsiter moves such that Reidemeister moves of type 3 are forbidden. In this paper we discuss colorings of virtual diagrams using an algebra, called a doodl…

Cactus doodles are geometric objects derived from cactus groups.

Paper defines doodles on closed surfaces, unifying classical and virtual theories.

Paper shows how to represent Milnor's triple linking number using chord diagrams and doodle invariants.

Computes invariants distinguishing between immersions and embeddings of doodles and blobs on surfaces.

In 1997 M.~Khovanov proved that any doodle can be presented as closure of twin, this result is analogue of classical Alexander's theorem for braids and links. We give a description of twins that have equivalent closures, this theorem is analogue of classical Markov theorem.

DOODL learns shared spectral dynamics across related dynamical systems.

To make music composition more approachable, we designed the first AI-powered Google Doodle, the Bach Doodle, where users can create their own melody and have it harmonized by a machine learning model Coconet (Huang et al., 2017) in the style of Bach. For users to input melodies, we designed a simplified sheet-music ba…

The paper classifies 10 antipodal pairings of self-dual maps.

Proves Alexander and Markov theorems for higher genus virtual doodles.

Given a plane curve $γ: S^1\to \mathbb R^2$, we consider the problem of determining the minimal number $I(γ)$ of inflections which curves $\mbox{diff}(γ)$ may have, where $\mbox{diff}$ runs over the group of diffeomorphisms of $\mathbb R^2$. We show that if $γ$ is an immersed curve with $D(γ)$ double points and no othe…

We investigate using reinforcement learning agents as generative models of images (extending arXiv:1804.01118). A generative agent controls a simulated painting environment, and is trained with rewards provided by a discriminator network simultaneously trained to assess the realism of the agent's samples, either uncond…

This paper explores Brunnian twin groups and their properties.