The knot I chose: "8_18". It is called 8_18 because it has 8 crossings and is the 18th knot on the knot table with 8 crossings. Picture of my knot (from http://katlas.math.toronto.edu/wiki/8_18) Crossing number: 8; the smallest number of crossings of any diagram with this knot is eight. Sequence of crossing switches that gives the unknot (upper bound on the unknotting number): An unknotting number is the number of changes you can make at intersections by changing which line is over or under at that intersection which at the end of all the changes gives you the unknot, or a simple loop with no knots in it. Basically you can change which line is over and which line is under at each intersection without sacrificing the integrity of the knot by cutting it in any way and the number of crossings you have to alter that give you the unknot becomes the upper bound of the unknotting number because there could be a simpler way to do it that requires less
Knots and crossing numbers A knot is a simple, closed, non-self-intersecting curve in \(R^3\). It is natural to think of a knot as constructed from a string glued together at the ends, usually tangled in the middle. The knot for my project is knot 9-1 in the knot atlas, which is centrosymmetric. The crossing number of the knot is 9. It is common to use a knot diagram or 2D projection to graphically represent knots. A knot is represented by a curved line on the page. When the knot passes under itself, a gap is used at the crossing. The projection of the 9-1 knot is shown in the following picture: Sequence of crossing switches for unknotting There are three types of simple allowed ways to deform knot diagrams by changing the number of crossings as shown in the following picture. The unknotting number of a knot is the minimum number of times to switch the crossing points to untie it. The 9-1 knot has the unknotting number \(4\). As shown in the fol