Non-alternating mappings

Construction of Non-alternating Knots

We investigate the behaviour of Rasmussen’s invariant s under the sharp operation on knots and obtain a lower bound for the sharp unknotting number. This bound leads us to an interesting move that transforms arbitrary knots into non-alternating knots.

Modified Halpern Iteration of Asymptotically Non-Expansive Mappings

Non-peripheral Ideal Decompositions of Alternating Knots

An ideal triangulation T of a hyperbolic 3-manifold M with one cusp is non-peripheral if no edge of T is homotopic to a curve in the boundary torus of M . For such a triangulation, the gluing and completeness equations can be solved to recover the hyperbolic structure of M . A planar projection of a knot gives four ideal cell decompositions of its complement (minus 2 balls), two of which are id...

Homologically Thin, Non-quasi-alternating Links

We exhibit the first examples of links which are homologically thin but not quasi-alternating. To show that they are not quasi-alternating, we argue that none of their branched double-covers bounds a negative definite 4-manifold with non-torsion H1. Using this method, we also complete the determination of the quasi-alternating pretzel links.

Compositions as Non-alternating Sequences of Partitions

Compositions are conceptualized as non alternating sequences of blocks of non-decreasing and strictly decreasing partitions. We find the generating function F (x, y, q) where x marks the size of the composition, y the number of parts and q the number of such partition blocks minus 1. We form these blocks starting on the left-hand-side of the composition and maximizing the size of each block. We...