  The Turaev genus defines a natural filtration on knots where Turaev genus
zero knots are precisely the alternating knots. We show that the signature of a
Turaev genus one knot is determined by the number of components in its all-A
Kauffman state, the number of positive crossings, and its determinant. We also
show that either the leading or trailing coefficient of the Jones polynomial of
a Turaev genus one link (or an almost alternating link) has absolute value one.
