TL;DR
This paper introduces the first-order genus, a new geometric invariant for knots in the three-sphere derived from gropes, and demonstrates its computability and implications for Seifert surface structures.
Contribution
It defines the first-order genus invariant, shows how to compute it for various knots, and explores its implications for the structure of Seifert surfaces.
Findings
The first-order genus can be computed for many knot examples.
It provides new insights into the structure of Seifert surfaces.
The invariant reveals geometric properties of knots related to gropes.
Abstract
We introduce a geometric invariant of knots in the three-sphere, called the first-order genus, that is derived from certain 2-complexes called gropes, and we show it is computable for many examples. While computing this invariant, we draw some interesting conclusions about the structure of a general Seifert surface for some knots.
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