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Minimum tetrahedral complexity for friends that are not 4-dimensional friends

Determine the minimum possible value of t(K) + t(K') among all pairs of non-isotopic hyperbolic knots K and K' with orientation-preservingly diffeomorphic 0-surgeries whose traces X(K) and X(K') are not orientation-preservingly homeomorphic (i.e., K and K' are friends but not 4-dimensional friends).

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Background

The paper studies pairs of knots (friends) whose 0-surgeries are orientation-preservingly diffeomorphic, and also the stronger notion of 4-dimensional friends where traces are diffeomorphic. Two complexity measures are investigated: crossing number and tetrahedral complexity t(K), the latter being the minimal number of tetrahedra in an ideal triangulation of the knot complement.

The authors determine the minimum of t(K)+t(K') among hyperbolic friends (Theorem 1.2), and they also determine the minimum of c(K)+c(K') among friends that are not 4-dimensional friends (Theorem 1.3). However, the analogous optimization for tetrahedral complexity in the non–4-dimensional-friends setting is left unresolved.

References

On the other hand, we can also ask for the simplest example of friends that are not $4$-dimensional friends. For the crossing number complexity, we can answer this question, while it remains open for the tetrahedral complexity.

— Complexity of equal 0-surgeries (2401.06015 - Abe et al., 11 Jan 2024) in Section 1 (Introduction), after Theorem 1.3