Research in Knot Theory: Khovanov Homology and 4-genus of Almost Alternating Knots
Mia DeStefano ’22, Wyatt Milgrim ’23, Ceci Villaseñor ’22, Emily Wadholm ’23, and Professor Adam Lowrance (Mathematics and Statistics)
Imagine you tie a string into a knot and glue the ends together: this is a mathematical knot in 3-space. When the 3-dimensional knot is projected into the plane, we obtain a picture of the knot called a knot diagram. We can look at each crossing of the knot and determine which strand goes over and which strand goes under in the diagram. Start at an arbitrary place in the knot and follow the string around. If it alternates between going over and under at each crossing, then we say the knot is alternating. Our work is on a specific class of knots, called almost alternating knots. These are knots that have a diagram where one can switch a single crossing from over to under to create an alternating diagram. There are certain properties of knots called invariants which help to differentiate between distinct knots. Our main result uses an algebraic knot invariant, called the Khovanov homology of the knot, to test whether the knot is almost alternating. We also examine the 4-genus of a knot—the fewest number of holes in any surface in four-space that is bounded by the knot. We prove that there are upper and lower bounds on the 4-genus of many almost alternating knots; these upper and lower bounds differ by one.