Topology Seminar: Gyo Taek Jin

Every knot can be presented on the union of finitely many half planes which have a common boundary line, so that each half plane contains a single arc of the knot. Such a presentation is called an arc presentation of the knot. The arc index of a knot is the minimal number of half planes needed in its arc presentations. A grid diagram of a knot is a knot diagram constructed by finitely many vertical line segments and the same number of horizontal line segments such that at each crossing a vertical segment crosses over a horizontal segment. A grid diagram with n vertical segments is easily converted to an arc presentation on n half planes, and vice versa. Grid diagrams are useful in several ways. A slight modification of a grid diagram gives a front projection of its Legendrian imbedding. Grid diagrams are used to compute Heegaard Floer homology and Khovanov homology. In this work, we've tabulated prime knots by arc index up to arc index twelve. This is achieved by generating grid diagrams of prime knots up to arc index twelve. This tabulation contains all prime alternating knots up to ten crossings and all prime non alternating knots up to twelve crossings. The largest crossing number among prime knots with arc index twelve is twenty eight.