Narrow Results

We consider here nonalternating knots and their properties. Specifically, we show certain classes of knots have nontrivial Jones polynomials.

In this paper, we compute the Khovanov homology over ℚ for (p, -p, q) pretzel knots for 3 ≤ p ≤ 15, p odd, and arbitrarily large q. We provide a conjecture for the general form of the Khovanov homology of (p, -p, q) pretzel knots. These computations reveal that these knots have thin Khovanov homology (over ℚ or ℤ). Because Greene has shown that these knots are not quasi-alternating, this provides an infinite class of non-quasi-alternating knots with thin Khovanov homology.

The 3-strand pretzel knots and links are a well-studied source of examples in knot theory. However, while there have been computations of the Khovanov homology and Rasmussen s-invariants of some sub-families of 3-strand pretzel knots, no general formula has been given for all of them. We give a formula for the unreduced Khovanov homology, over the rational numbers, of all 3-strand pretzel links. We also compute generalized s-invariants of these links.