Saturday, September 29, 2012
Reed Caron (B.S. 2013, Grinnell College) presents The Jones Polynominal of Spherical Virtual Knots today at Grinnell College.
The concept of a virtual knot was first discovered by Louis Kauffman in 1999. A virtual knot is a knot whose diagram possesses virtual crossings along with the classical over/under crossings A virtual crossing is denoted by an intersection of two strands enclosed by a circle. Two virtual knots are considered to be equivalent if one can be turned into the other by performing a combination of the seven "Reidemaster Moves." A spherical virtual knot is a knot whose diagram can be represented by a string traveling around a surface of a sphere. As the string crosses itself on the sphere, virtual crossings are created. Classical crossings are represented by a line of string on the northern hemisphere mirroring a line of string on the southern hemisphere. (The string on the northern hemisphere represents the overcrossing while the string on the southern hemisphere represents the undercrossing,) To illustrate the spherical representation of a knot, we draw a circle to represent the sphere's equator and then depict the parts of the string on the northern hemisphere as solid lines and the parts on the southern hemisphere as dotted lines. The Jones Polynominal, discussed in 1984 by Vaughan Jones, is a knot invariant under the aforementioned Reidemaster Moves. However, figure 4 displays a particular move that looks similar to a Reidemaster Move, yet does in fact change a knot's Jones Polynominal. This move is referenced to by knot theorists as the Forbidden Move.
Every spherical virtual knot has a symmetric Jones Polynominal. For example, according to the conjecture, the Jones Polynominal of a spherical virtual knot could be [formula]. In this poster, we will show that the conjecture is true for a specific family of spherical virtual knots.