Diagrammatic and Geometric Invariants of Hyperbolic Weakly Generalized Alternating Knots

We study the relationship between knot and link diagrams on surfaces and their invariants coming from hyperbolic geometry. A \indef{link diagram on a surface}, denoted $\pi(L)\subset F$, is a way of projecting a link $L$ in some manifold $M$ onto some surface $F\subset M$. This generalizes the notion of a link diagram, and has been studied with a variety of conditions. We will work with the weakly generalized alternating knots and links of Howie and Purcell, which gives certain criteria to ensure the diagram is interesting. A cusp of a hyperbolic knot $K$ is a neighborhood of $K$ in $M\setminus K$, and the cusp volume is the Euclidean volume of a maximal cusp. We show that the cusp volume of a weakly generalized alternating knot, with some additional conditions, is bounded both above and below based on the twist number of $\pi(K)$ and $\chi(F)$. This is done by constructing a new essential surface for $K$ that has nice properties, including having Euler characteristic based on the twist number as opposed to the crossing number. Our bound on cusp volume leads to interesting bounds on other geometric properties of $K$, including slope length and volumes of Dehn surgery. The volume}of a hyperbolic link $L$ in a manifold $M$ is the hyperbolic volume of the complement $M\setminus L$. We can show that volume is also bounded below by the twist number of $\pi(L)$ and $\chi(F)$. We do this by generalizing the Jones polynomial to weakly generalized alternating links, and showing that there is a relation between this polynomial and the twist number, and this polynomial and the volume. Through the course of proving this bound, we also get relations between the guts of the checkerboard surfaces of $L$ and this generalized Jones polynomial. In addition, if we are working inside a thickened surface, the twist number becomes an isotopy invariant of $L$.