Fabiola Manjarrez-Gutiérrez

Centro de Investigación en Matemáticas, A.C.,
Jalisco S/N, Col. Valenciana,
Guanajuato, Gto. 36240, MEXICO.

Title: Non-fibered free genus one knots are almost fibered

Abstract: A circle-valued Morse function on the exterior of a knot induces a circular handle decomposition on the knot exterior, in such a way that the regular levels contain Seifert surfaces for the knot. We can encode the complexity of these surfaces in an n-tuple called the circular width of the knot exterior, $cw(E(K))$. The circular handle decomposition that realizes $cw(E(K))$ is called circular thin position of $E(K)$. A knot is almost fibered if there is a circular thin position realizing $cw(K)=(m)$, in other words there is a circular thin position containing exactly two regular levels. In joint work with V. Núñez and E. Ramírez-Losada, we develop practical techniques to construct circular decompositions of knots with free Seifert surfaces in the 3-sphere, and compute handle numbers of many knots. We also show that all non-fibered free genus one knots are almost fibered.