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We examine the conjecture, due to Champanerkar, Kofman, and Purcell that$\text{vol}(K) < 2 \pi \log \det (K)$ for alternating hyperbolic links, where$\text{vol}(K) = \text{vol}(S^3\backslash K)$ is the hyperbolic volume and$\det(K)$ is the determinant of $K$. We prove that the conjecture holds for2-bridge links, alternating 3-braids, and various other infinite families. Weshow the conjecture holds for highly twisted links and quantify this by showingthe conjecture holds when the crossing number of $K$ exceeds some function ofthe twist number of $K$.

The Determinant and Volume of 2-Bridge Links and Alternating 3-Braids