Average Genus of Oriented Rational Links
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Abstract
The goal of this dissertation is to develop the formulation of the average minimal genus of all reduced alternating rational links with a given crossing number. Work has been done by N. Dunfield to approximate the growth of the genus of knots with respect to the crossing number. It has been hypothesized that the growth is linear. M. Cohen has recently submitted a paper providing a lower bound estimate of the average genus of knots with a given crossing number \cite{cohen2021lower}. However, the actual average for knots has remained elusive and the average genus of links even more so. Using the counting methods in a paper produced by Y. Diao, M. Finney, and D. Ray \cite{diao2021number}, we are able to derive a precise average for the genus of links and knots with crossing number given and a weighted average of the minimal genus for all rational links.This dissertation consists of six chapters. Our most significant contribution and calculations are presented in chapters four and five after we familiarize the reader on the essentials of knot theory and its invariants and review the results from our paper on enumerating the rational links. The structure of the dissertation is organized as follows. In the first chapter, we provide a background of the field of knot theory as it relates to our results. The second chapter consists of detailed definitions and will familiarize the reader with the field of knot theory. The next chapter will present the results from our enumeration paper including examples on how we addressed over counting and a systematic walk-through of the construction of our computations. The fourth chapter will give specifics on how we were able to count the number of rational links given a crossing number and the number of Seifert circles in the decomposition. In the fifth chapter we will discuss the theorems that will lay the final piece needed to determine the average genus and then discuss the final results. Lastly, the sixth chapter will discuss another type of link, the Montesinos link, and details how the results of our paper can be extended to future work in enumerating this special class of links.