The maximum volume of hyperbolic polyhedra

Turaev-Viro invariants and cabling operations

“…By the same argument, each iteration of Step (2) contributes a nonzero factor to the determinant and clears one of the two nonzero entries two rows up. Since i − is odd, all rows cleared have odd index in R, and Step (2) terminates when we expand along row (1).…”

“…• If m + 3 ≤ j < r, then f + l = e r−j = e r−j+2 = f − l . This implies that the row indices of the nonzero entries in each column differ by at most two for every column except col (1).…”

“…Let i − = q 2 − 1 and i + = q 2 + 1. Each case uses a nearly identical argument, so we will limit ourselves to Case (1). Here,…”

“…4. Some examples which satisfy the hypothesis of Corollary 1.5 include the figure-eight knot and the Borromean rings by Detcherry-Kalfagianni-Yang [8], the Whitehead chains by Wong [25], the fundamental shadow links by Belletti-Detcherry-Kalfagianni-Yang [2], a family of hyperbolic links in S 2 × S 1 by Belletti [1], a large family of octahedral links in S 3 by the first author of this paper [13], and a family of link complements in trivial S 1 -bundles over oriented connected closed surfaces by the authors of this paper [14].…”

On volumes of hyperbolic right-angled polyhedra

“…An initial list of ideal right-angled polyhedra is given in [EV20c], and of compact ones in [Ino19]. It is worth mentioning that recently studying volumes of ideal right-angled polyhedra has become more attractive and important in view of the problem of maximal volumes [Bel21] and the conjecture about hyperbolic right-angled knots [CKP19]. A detailed discussion of constructions of hyperbolic 3-manifolds from right-angled polyhedra can be found in the recent survey [Ves17].…”