Complexity of equal 0-surgeries

Tetsuya Abe; Marc Kegel; Nicolas Weiss

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arXiv · 2401.06015

Complexity of equal 0-surgeries

Tetsuya Abe·Marc Kegel·Nicolas Weiss

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Abstract

We say that two knots are friends if they share the same 0-surgery. Two friends with different sliceness status would provide a counterexample to the 4-dimensional smooth Poincar\'e conjecture. Here we create a census of all friends with small crossing numbers c and tetrahedral complexities t, and compute their smooth 4-genera. In particular, we compute the minimum of c(K)+c(K') and of t(K)+t(K') among all friends K and K'. Along the way, we classify all 0-surgeries of prime knots of at most 15 crossings. Moreover, we determine for many friends in our census if their traces are equivalent or not. For that, we develop a new obstruction for two traces being homeomorphic coming from symmetry-exceptional slopes of hyperbolic knots. This is enough to also determine the minimum value of c(K)+c(K') among all friends K and K' whose traces are not homeomorphic.

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BibTeX RIS

Tetsuya Abe, Marc Kegel, Nicolas Weiss. 2024-01-11. Complexity of equal 0-surgeries. https://arxiv.org/abs/2401.06015

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