Labeled binary planar trees and quasi-Lie algebras

The Lie algebra of rooted planar trees

We study a natural Lie algebra structure on the free vector space generated by all rooted planar trees as the associated Lie algebra of the nonsymmetric operad (non-$Σ$ operad, preoperad) of rooted planar trees. We determine whether the Lie algebra and some related Lie algebras are finitely generated or not, and prove that a natural surjection called the augmentation homomorphism onto the Lie algebra of polynomial vector fields on the line has no splitting preserving the units.

球面纽结类的辫状等价性

A braidoid equivalence for spherical knotoids

本研究针对球面纽结类的辫状表示问题，通过细化辫状图的L-等价概念，建立了S²中（多）纽结类图以带标记辫状图闭包表示时的等价定理，完善了辫状理论与纽结类理论的对应关系。

Toroidal embeddings of abstractly planar graphs are knotted or linked

We give explicit deformations of embeddings of abstractly planar graphs that lie on the standard torus $T^2 \subset \mathbb{R}^3$ and that contain neither a nontrivial knot nor a nonsplit link into the plane. It follows that ravels do not embed on the torus. Our results provide general insight into properties of molecules that are synthesized on a torus.

Combinatorics of injective words for Temperley-Lieb algebras

This paper studies combinatorial properties of the 'complex of planar injective words', a chain complex of modules over the Temperley-Lieb algebra that arose in our work on homological stability. Despite being a linear rather than a discrete object, our chain complex nevertheless exhibits interesting combinatorial properties. We show that the Euler characteristic of this complex is the n-th Fine number. We obtain an alternating sum formula for the representation given by its top-dimensional homology module and, under further restrictions on the ground ring, we decompose this module in terms of…

Planar algebras and the decategorification of bordered Khovanov homology

We give a simple, combinatorial construction of a unital, spherical, non-degenerate $\ast$-planar algebra over the ring $\mathbb{Z}[q^{1/2},q^{-1/2}]$. This planar algebra is similar in spirit to the Temperley-Lieb planar algebra, but computations show that they are different. The construction comes from the combinatorics of the decategorifications of the type A and type D structures in the author's previous work on bordered Khovanov homology. In particular, the construction illustrates how gluing of tangles occurs in his bordered Khovanov homology and its difference from that in Khovanov's ta…