On the kernel of the surgery map restricted to the 1-loop part

TL;DR

This paper investigates the kernel of the surgery map on the 1-loop part of Jacobi diagrams, revealing its structure and introducing refined relations among claspers to better understand the algebraic properties of homology cylinders.

Contribution

It precisely determines the kernel of the surgery map restricted to the 1-loop part after a specific quotient, and introduces refined AS and STU relations among claspers.

Findings

Kernel of the surgery map on 1-loop part is explicitly characterized.

Refined AS and STU relations among claspers are proposed.

Structure of the abelian group $Y_n\mathcal{IC}_{g,1}/Y_{n+2}$ is studied.

Abstract

Every homology cylinder is obtained from Jacobi diagrams by clasper surgery. The surgery map $\mathfrak{s} \colon \mathcal{A}_n^c \to Y_n\mathcal{IC}_{g,1}/Y_{n+1}$ is surjective for $n \geq 2$, and its kernel is closely related to the symmetry of Jacobi diagrams. We determine the kernel of $\mathfrak{s}$ restricted to the 1-loop part after taking a certain quotient of the target. Also, we introduce refined versions of the AS and STU relations among claspers and study the abelian group $Y_n\mathcal{IC}_{g,1}/Y_{n+2}$ for $n \geq 2$.