Equality conjecture for the complexity bounds of the 42 magic-manifold knot families

Let $\{K(i)\}_{i\in\mathbb{Z}}$ be the infinite extension of one of the 42 distinguished families of census knots, with breadth $x$, and let $c(K)$ denote the complexity of the knot. For each family, the constructed triangulations give, for $j>0$, the bound

$$c\big(K(-x-j)\big)=c\big(K(x-1+j)\big)\leq 9+j.$$

Complexity equality conjecture. The inequality in the bounded-complexity result can be upgraded to equality for all 42 families of knots in this paper.

The equality would establish that the constructed triangulations remain minimal throughout all 42 infinite families, extending the confirmed minimality of the census knots beyond the currently verified cases.