KHOVANOV HOMOLOGY AND THE TWIST NUMBER OF ALTERNATING KNOTS

In A Volumish Theorem for the Jones Polynomial, by O. Dasbach and X. S. Lin, it was shown that the sum of the absolute value of the second and penultimate coefficient of the Jones polynomial of an alternating knot is equal to the twist number of the knot. Here we give a new proof of this result using Khovanov homology. The proof is by induction on the number of crossings using the long exact sequence in Khovanov homology corresponding to the Kauffman bracket skein relation.