Manolescu and Piccirillo (2023) recently initiated a program to construct an exotic or by using zero surgery homeomorphisms and Rasmussen's ‐invariant. They find five knots that if any were slice, one could construct an exotic and disprove the Smooth 4‐dimensional Poincaré conjecture. We rule out this exciting possibility and show that these knots are not slice. To do this, we use a zero surgery homeomorphism to relate slice properties of two knots
Jones diameter and crossing numbers of knots
It has long been known that the quadratic term in the degree of the colored Jones polynomial of a knot is bounded above in terms of the crossing number of the knot. We show that this bound is sharp if and only if the knot is adequate.
As an application of our result we determine the crossing numbers of broad families of non-adequate prime satellite knots. More specifically, we exhibit minimal crossing number diagrams for untwisted Whitehead doubles of zero-writhe adequate knots. This allows us to determine the crossing number of untwisted Whitehead doubles of any amphicheiral adequate knot, including, for instance, the Whitehead doubles of the connected sum of any alternating knot with its mirror image.
We also determine the crossing number of the connected sum of any adequate knot with an untwisted Whitehead double of a zero-writhe adequate knot.
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- Award ID(s):
- 2004155
- NSF-PAR ID:
- 10417820
- Date Published:
- Journal Name:
- Advances in Mathematics
- Volume:
- 417
- ISSN:
- 1857-8438
- Page Range / eLocation ID:
- 108937
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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