# Winding Number of $r$-modular sequences and Applications to the Singularity Content of a Fano Polygon

## TL;DR

This paper generalizes the winding number concept for lattice point sequences and applies it to classify certain Fano polygons based on their singularity content, providing new restrictions for their structure.

## Contribution

It introduces a generalized winding number formula for lattice sequences and uses it to classify Fano polygons with specific singularity configurations.

## Key findings

- Classified Fano polygons without T-singularities using winding number restrictions.
- Derived conditions on residual singularities of Fano polygons.
- Provided a new tool for analyzing the structure of Fano polygons.

## Abstract

By generalising the notion of a unimodular sequence, we create an expression for the winding number of certain ordered sets of lattice points. Since the winding number of the vertices of a Fano polygon is necessarily one, we use this expression as a restriction to classify all Fano polygons without T-singularities and whose basket of residual singularities is of the form $\left\{ \frac{1}{r}(1,s_{1}), \frac{1}{r}(1,s_{2}), \ldots, \frac{1}{r}(1,s_{k}) \right\}$ for $k,r \in \mathbb{Z}_{>0}$, and $1 \leq s_{i} < r$ is coprime to $r$.

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Source: https://tomesphere.com/paper/1901.10929