On unknotting numbers and knot trivadjacency

Abstract

We prove for rational knots a conjecture of Adams et al. that an alternating unknotting number one knot has an alternating unknotting number one diagram. We use this then to show a refined signed version of the Kanenobu-Murakami theorem on unknotting number one rational knots. Together with a similar refinement of the linking form condition of Montesinos-Lickorish and the HOMFLY polynomial, we prove a condition for a knot to be $2$-trivadjacent, improving the previously known condition on the degree-2-Vassiliev invariant. We finally show several partial cases of the conjecture that the knots with everywhere $1$-trivial knot diagrams are exactly the trivial, trefoil and figure eight knots. (A knot diagram is called everywhere $n$-trivial, if it turns into an unknot diagram by switching any set of $n$ of its crossings.)