"It is known that the Slope Conjecture is true for an adequate link, and that the colored Jones polynomial of a semi-adequate link has a well-defined tail (head) consisting of stable coefficients, which carry geometric and topological information of the link complement. We study the colored Jones polynomial of a link that is not semi-adequate and show that a tail (head) consisting of stable coefficients of the polynomial can also be defined. Then, we prove the Slope Conjecture for a new family of pretzel knots which are not adequate. We also study the relationship between the Jones polynomial and the topology of the knot complement by relating its coefficients to the non-orientable genus of an alternating knot. The two-sided bound we obtain can often determine the non-orientable genus. Lastly, we develop the connection between the Jones polynomial and a knot invariant coming from Heegaard Floer homology by using it to classify 3-braids which are L-space knots." -- Abstract.
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