An Algebraic Matching Algorithm for the Genus of Grid Diagrams

Abstract

Grid diagrams encode knots and links combinatorially through an (n,n) winding matrix W(G). A fundamental invariant of the grid diagram is its genus g(G), which records essential topological information about the underlying knot. We give an algebraic algorithm that computes g(G) directly from W(G). The method repurposes the Hungarian algorithm ,after a column normalization that makes the matrix non-negative, it applies row-column reductions and minimum-vertex-cover corrections until the zero entries admit a perfect matching, equivalently until the minimum marking number equals n. By Konig's theorem the resulting permutation defines a distinguished grid state, and g(G) is then read from this matrix via the classical grid homology formula (Ozsvath-Stipsicz-Szabo). The novelty of this paper is the algorithm and its strongly polynomial analysis; the underlying invariant is known grid-homology theory. We illustrate the method on the trefoil and additional examples, and discuss extensions to (m,n) matrices and grid-move navigation toward the knot genus g(K).