Graphs, determinants of knots and hyperbolic volume

Mahler measure and the Vol‐Det Conjecture

“…Experimental evidence has long suggested a close relationship between the volume and determinant of alternating knots [23,37]. The following inequality was conjectured in [15], and verified for all alternating knots up to 16 crossings, weaving knots [16] with hundreds of crossings, all 2-bridge links and alternating closed 3-braids [11].…”

Geometrically and diagrammatically maximal knots

“…The top nine knots in this census sorted by maximum volume and by maximum determinant agree, but only set-wise! More data and a broader context is provided by Friedl and Jackson [14], and Stoimenow [29]. In particular, Stoimenow [29] proved there are constants C 1 , C 2 > 0, such that for any hyperbolic alternating link K,…”

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

“…The determinant of a link K is defined by det(K) = |∆ K (−1)| where ∆ K (t) is the Alexander polynomial. It is well-known when K is alternating, the determinant is equal to the number of spanning trees of any of the checkerboard graphs for K (see for example [16,Lemma 3.14]). Recall that the checkerboard graphs for K are constructed as follows.…”

“…Stoimenow [16] also explored the relationship between volume and determinant, and showed that if K is a non-trivial, non-split, alternating hyperbolic link then (1.1) det(K) ≥ 2(1.0355) vol (K) He further demonstrated that there exist constants C 1 , C 2 > 0 such that for any hyperbolic link K…”

“…We rely on a recurrence equation due to Kauffman [10] as well as two well-known combinatorial theorems for counting spanning trees. In the case of highly twisted knots, we utilize work of Stoimenow [16] who provided a lower bound on the number of spanning trees in certain graphs.…”