Gabriel Doyle '05
Department of Mathematics
This picture is a representation of the universal cover of the doubly-pointed Heegaard diagram of genus 1 of a (1,1)-knot. The black line represents the bounding curve for the knot, and the gray lines represent a meridian and a longitude of the torus. By finding all disks bounded by a vertical segment of the gray lines and any segment of the black lines, one is able to calculate the Knot Floer Homology of a (1,1)-knot. This is a knot invariant that can be used to tell similar knots apart. All 22 disks that can be used to determine the Floer Homology of this knot are marked on this diagram, with green and blue referring to the multiplicities of two special points. Bright colors indicate disks with multiplicity 1, while the dark disks have multiplicity 2. The title of the diagram is a set of four integers that define this particular knot: the number of intersections between the black and grey line on one side of the fundamental region (one small square in the picture), the number of disks on one side of the fundamental region, the number of lines going above the right-hand side’s disks, and a rotation number.