# 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.