EulerKnicklast

EulerKnicklast is a fictional topological invariant used in mathematical folklore and puzzle communities to study how boundary shape interacts with topology. It is not a standard invariant but serves as a teaching tool to illustrate boundary effects.

Definition: For a compact 2-dimensional manifold with boundary X that is polygonally decomposed, the EulerKnicklast EKL(X) is defined as EKL(X) = χ(X) + N(X), where χ(X) is the Euler characteristic and N(X) is the knick count. The knick count equals the number of boundary vertices at which the boundary angle is concave (interior angle greater than 180 degrees). For smooth boundaries, N(X) is obtained by polygonalizing the boundary and counting concave vertices, or via an appropriate limiting procedure.

Properties: EKL is invariant under boundary-preserving homeomorphisms, and EKL adds under disjoint unions. If the boundary contains no concave features (N(X) = 0), EKL(X) = χ(X). Because it depends on a boundary discretization, EKL is not a pure topological invariant and is used descriptively rather than as a formal axiom.

Examples: A disk has χ = 1 and N = 0, so EKL = 1. A simple convex polygon also has EKL = 1. A polygon with a single inward notch (two concave corners) has χ = 1 and N = 2, so EKL = 3.

History and usage: The term arose in online puzzle communities and classroom contexts to provoke discussion about what happens when you modify a boundary. It is occasionally used as a mnemonic aid or thought experiment.

See also: Euler characteristic, topology, boundary, polygonal decomposition.