Crosscap numbers and the Jones polynomial
Efstratia Kalfagianni, Christine Ruey Shan Lee
TL;DR
This paper establishes precise bounds relating the crosscap number of alternating links to their Jones polynomial, enabling exact calculations for various families and exploring extensions to non-alternating links.
Contribution
It provides the first sharp two-sided bounds connecting crosscap number and Jones polynomial for alternating links, with applications to specific families and non-alternating links.
Findings
Sharp bounds for crosscap number in terms of Jones polynomial
Exact crosscap numbers for several infinite families of links
Extensions to non-alternating links discussed
Abstract
We give sharp two-sided linear bounds of the crosscap number (non-orientable genus) of alternating links in terms of their Jones polynomial. Our estimates are often exact and we use them to calculate the crosscap numbers for several infinite families of alternating links and for several alternating knots with up to twelve crossings. We also discuss generalizations of our results for classes of non-alternating links.
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