Khaled Qazaqzeh

Characterization of quasi-alternating Montesinos links

Let L be a quasi-alternating link at a crossing c. We construct an infinite family of quasi-alternating links from L by replacing the crossing c by a product of rational tangles, each of which extends c. Consequently, we determine an infinite family of quasi-alternating Montesinos links. This family includes all classes of quasi-alternating Montesinos links that have been detected by Widmer [Quasi-alternating Montesinos links, J. Knot Theory Ramifications18(10) (2009) 1459–1469]. We conjecture that this family contains all non-alternating quasi-alternating Montesinos links.

The parity of the Maslov index and the even cobordism category

We give a formula for the parity of the Maslov index of a triple of Lagrangian subspaces of a skew symmetric bilinear form over the real numbers. We define an index two subcategory (the even subcategory) of a 3-dimensional cobordism category. The objects of the category are surfaces are equipped with Lagrangian subspaces of their real first homology. This generalizes a result of the first author where surfaces are equipped with Lagrangian subspaces of their rational first homology.

A REMARK ON THE DETERMINANT OF QUASI-ALTERNATING LINKS

We show that the crossing number of any link that is known to be quasi-alternating is less than or equal to its determinant. Based on this, we conjecture that the crossing number of any quasi-alternating link is less than or equal to its determinant. Thus if this conjecture is true, then it gives a new property of quasi-alternating links and easy obstruction to a link being quasi-alternating.

Integral bases for certain TQFT-modules of the torus

We find two bases for the lattices of the SU(2)-TQFT-theory modules of the torus over given rings of integers. One basis is a variation on the bases defined in [GMW] for the lattices of the SO(3)-TQFT-theory modules of the torus. Moreover, we discuss the quantization functors (Vp, Zp) for p = 1, and p = 2. Then we give concrete bases for the lattices of the modules in the 2-theory. We use the above results to discuss the ideal invariant defined in [FK]. The ideal can be computed for all the 3-manifolds using the 2-theory, and for all 3-manifolds with torus boundary using the SU(2)-TQFT-theory. In fact, we show that this ideal using the SU(2)-TQFT-theory is contained in the product of the ideals using the 2-theory and the SO(3)-TQFT-theory under a certain change of coefficients, and with equality in the case of torus boundary.