Knots with special properties

Mario Eudave Muñoz (IMATE - Juriquilla)


(See original in Spanish )

Let $K$ be a knot. The unknotting number of $K$, $u(K)$, is the minimum number of crossing changes required to unknot K. A knot can also be unknotted using bands. Let $u_b(K)$ be the minimum number of bands we need to put on $K$ to obtain the unknot.

It is easy to construct a knot with $u(K) = 1$, but if we want $u(K)=1$ plus some other property it can be really hard or impossible to construct. For example, there are no composite knots with $u(K)=1$. There are some satellite knots or knots with conway spheres with $u(K)=1$, but they are rare. Something similar happens for knots with $u_b(K)=1$.

This talk will be about the construction of knots with $u(K)=1$ or $u_b(K)=1$ plus some special properties.

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