**Project leader:** Professor **Calvin McPhail-Snyder**

**Project manager: **Jason Ma

**Team members: **Rebecca Lan, Khiyali Pillalamarri, Lorenzo Valerio, Wendy Wang

Take a string and tie it into a knot, then put the knot in hyperbolic space. The knot complement is all of the space around the knot, excluding the knot itself. Our project aims to understand the knot complement by dividing it up into tetrahedra. We make this problem algebraic by drawing a diagram of the knot, then assigning three numbers *(a, b, m)* to each segment of the diagram. For the tetrahedra to glue together correctly the various *(a, b, m)* must satisfy certain equations. Our project explores these equations, particularly their application to links. Links are like knots, except that they allow for multiple strings to be tied together. This makes the equations more difficult to solve, because for a knot *m* is the same everywhere, but for a link there is a different *m* for every string.

Starting from hyperbolic structures on link complements described by octahedral decomposition, we considered the ideal triangulation to diagrams of the links. The decompositions gave sets of gluing equations from a segment or a region of the knots. We calculated through the parallel, antiparallel, negative-crossing, and positive-crossing twisted regions, and simplified the relations by alternating *W*-Fibonacci sequences. Later, we applied the relations to a more general case of gluing various twisted regions together. Examples included torus knot with odd and even number of crossings. To generalize it, we delved into the gluing equations for a positive-crossing region glued to a negative-crossing region which gave many interesting knots including the figure-8 knot. We also used our methods to compute the *A*-polynomials of these knots and links.