# Universal Surgery Problems with Trivial Lagrangian

## TL;DR

This paper investigates how certain geometric moves affect boundary links in 4-dimensional surgery, demonstrating that universal links admit Seifert surfaces with trivial Lagrangian and proposing conditions for sliceness.

## Contribution

It introduces a new class of boundary links with specific Seifert matrix properties and corrects a Kirby calculus identity relevant to link-slice problems.

## Key findings

- Universal links admit Seifert surfaces with trivial Lagrangian.
- A restrictive Seifert matrix condition implies links are slice.
- Corrected Kirby calculus identity aids in surgery kernel construction.

## Abstract

We study the effect of Nielsen moves and their geometric counterparts, handle slides, on good boundary links. A collection of links, universal for 4-dimensional surgery, is shown to admit Seifert surfaces with trivial Lagrangian. They are good boundary links, with Seifert matrices of a more general form than in known constructions of slice links. We show that a certain more restrictive condition on Seifert matrices is sufficient for proving the links are slice. We also give a correction of a Kirby calculus identity in \cite{FK2}, useful for constructing surgery kernels associated to link-slice problems.