A knot is an embedding of S1 in three-dimensional space. Generally, it can be thought of as a knotted piece of string with the ends glued together. When we project a knot into the plane, we can create a knot diagram in which we specify which portion of the string lies on top at each place that the string crosses itself. To perform a crossing change on a knot, one can imagine cutting one portion of the string at a crossing, allowing another portion of the string to pass through, and then gluing the cleaved ends back together. We define the distance between two knots, K1 and K2, to be the minimum number of crossing changes one must perform on either K1 or K2 to obtain the other knot.
Circular DNA can become knotted during biological processes such as recombination and replication. We can model knotted DNA with a mathematical knot. Type II topoisomerases are the enzymes tasked with keeping DNA unknotted, and they act on double-stranded circular DNA by breaking the backbone of the DNA, allowing another segment of DNA to pass through, and then re-sealing the break. Thus, performing a crossing change on a knot models the action of this protein. Specifically, studying knots of distance one can help us better understand how the action of a type II topisomerase on double-stranded circular DNA can alter DNA topology.
We create a knot distance graph by letting the set of vertices be rational knots with up to and including thirteen crossings and by placing an edge between two vertices if the two knots corresponding to those vertices are of distance one. A neighborhood of a vertex, v, in a graph is the set of vertices with which v is adjacent via an edge. Using graph theoretical and topological tools, we examine graphs of knot distances and define a mapping between distance one neighborhoods. Additionally, this idea can also be examined and visualized as performing Dehn surgery on the double branched cover of a knot.