A geometric proof of the Milnor conjecture

TL;DR

This paper provides a geometric proof that the Seifert genus equals the slice genus for positive knots and introduces a new combinatorial invariant to estimate the slice genus for all knots.

Contribution

It offers a geometric proof for a key conjecture and introduces a novel combinatorial invariant for bounding the slice genus of knots.

Findings

Seifert genus equals slice genus for positive knots

A new combinatorial invariant bounds the slice genus

Properties and applications of the invariant are discussed

Abstract

A geometric argument is given to prove that the Seifert genus of a positive knot equals its slice genus. A combinatorial invariant, giving a lower bound for the slice genus, is formulated for arbitrary knots. Properties and applications of this invariant are discussed.