The quadratic linking degree

TL;DR

This paper introduces a quadratic linking degree in algebraic geometry using motivic homotopy theory, providing a new invariant analogous to linking numbers that takes values in the Witt group, with explicit computation methods and examples.

Contribution

It develops a quadratic linking degree as an algebraic analogue of linking numbers, extending motivic homotopy theory to algebraic links and providing explicit computation techniques.

Findings

Defined the quadratic linking degree in algebraic geometry.

Established properties of the quadratic linking degree.

Provided explicit computations for algebraic analogues of torus links.

Abstract

By using motivic homotopy theory, we introduce a counterpart in algebraic geometry to oriented links and their linking numbers. After constructing the (ambient) quadratic linking degree -- our analogue of the linking number which takes values in the Witt group of the ground field -- and exploring some of its properties, we give a method to explicitly compute it. We illustrate this method on a family of examples which are analogues of torus links, in particular of the Hopf and Solomon links.