Polyomino and Rectilinear Knot Projections

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, NY 11451

February 6, 2018

Knots in 3-dimensions can be projected onto a plane and studied as 4-valent plane graphs. By adding 2-valent vertices one can represent knots as rectilinear polygons and/or polyominoes. While 4-valent connected plane graphs have Eulerian circuits, only knot projections have the property of having "cut-through" Eulerian circuits. Interesting mathematical questions arise from studying parameters of such plane graphs, for example, the number of sides of the infinite face, number of faces with i-sides, number of vertices of valence j, and numbers of sides of polygons which can be associated with the knot drawings.