Narrow Results

A classical result states that the determinant of an alternating link is equal to the number of spanning trees in a checkerboard graph of an alternating connected projection of the link. We generalize this result to show that the determinant is the alternating sum of the number of quasi-trees of genus j of the dessin of a non-alternating link.

Furthermore, we obtain formulas for coefficients of the Jones polynomial by counting quantities on dessins. In particular, we will show that the jth coefficient of the Jones polynomial is given by sub-dessins of genus less or equal to j.

We give an upper bound for the dealternating number of a closed 3-braid. As applications, we determine the dealternating numbers, the alternation numbers and the Turaev genera of some closed positive 3-braids. We also show that there exist infinitely many positive knots with any dealternating number (or any alternation number) and any braid index.

We describe a correspondence between Turaev surfaces of link diagrams on S² ⊂ S³ and special Heegaard diagrams for S³ adapted to links.

For each positive integer $n$ we will construct a family of infinitely many hyperbolic prime knots with alternation number 1, dealternating number equal to $n$, braid index equal to $n+3$ and Turaev genus equal to $n$.

The Turaev genus and dealternating number of a link are two invariants that measure how far away a link is from alternating. We determine the Turaev genus of a torus knot with five or fewer strands either exactly or up to an error of at most one. We also determine the dealternating number of a torus knot with five or fewer strands up to an error of at most two. We also give bounds on the Turaev genus and dealternating number of torus links with five or fewer strands and on some infinite families of torus links on six strands.