Bridge presentations of knotted objects

Shapes such as circles and spheres can be knotted up in very complicated ways. This thesis uses tools from algebra and topology to measure how complexly knotted a given shape is. Successfully doing so can lead to a better understanding of knot-like biological structures such as proteins and DNA.

A common method of studying a knotted object is to break it up into simpler pieces, and then analyze how the pieces fit back together. It is well known that a knotted circle (resp. a knotted sphere) can always be decomposed into two collections of untied segments (resp. three collections of untied disks). Such a decomposition is called a bridge decomposition. Results in this thesis provide an efficient way to compute the minimum number of untied segments and disks needed for a bridge decomposition of the knotted object. If such a number is large, then one can conclude that the object is highly entangled.

Given a random picture of a knotted object, there is a straightforward method to obtain a bridge decomposition that is usually made up of too many untied segments and disks. By results in this thesis, one can often obtain a more efficient bridge decomposition by searching for a particular way to color the knotted objects. This simple yet effective coloring technique produces optimal bridge decompositions for various knotted objects in three-dimensional and four-dimensional spaces.