Knot Theory

Dr. Jennifer McLoud-Mann, Faculty Mentor


Knot theory is very accessible to undergraduate students, and Dr. McLoud has successfully mentored many undergraduate research projects in this area. Student authors have presented their work at regional/state meetings, and many have published manuscripts.

Several problems related to Dr. McLoud’s research interests remain open for investigation:

math pic Cell Knots. Given a lattice L, a lattice knot K is a closed nonintersecting polygonal path that consists of unit edges whose endpoints lie on the lattice. The minimum edge number of a knot K is the smallest number of unit sticks that can be used to form the knot K, denoted mK. Minimum edge numbers for trefoil knots have been found in many standard lattices. A natural Voronoi tessellation of space is associated with a given lattice, and the 3D Voronoi cells that tesselate can be used to construct a cell knot. The minimum number of cells to form a knot K is called the minimum cell number of K and is denoted MK. The connection between mK and MK for a given knot K can be studied. For the simple cubic lattice, MK = 2 mk. Preliminary results indicate that the same result holds for the simple hexagonal lattice. Exploring the connection between mK and MK in the face-centered cubic lattice and the body-centered cubic lattice, important to chemists, will be interesting and valuable.

Sphere Number for Small Knots. Given a lattice L, the ball centered at the origin of radius r, BL(r), is dened to be all points of the lattice at a distance r or less from the origin. Some research questions include

1) what is the smallest radius of a ball containing a nontrivial knot.

2) what is the smallest radius of a ball containing a knot K

3) how do the answers to previous questions relate to edge number and stick number.

4) how do these numbers relate to curvature and writhe? These questions can be explored in the simple cubic,
simple hexagonal, face-centered cubic, and body-centered cubic lattices with the aid of the Hyak computing cluster.



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