The Unknotting number and band-unknotting number of a knot

Abstract

We show some results on the unknotting number and the band-unknotting number. Taniyama characterized knots whose unknotting number is half the crossing number minus one. We show that if the unknotting number of a knot is half the crossing number minus two, then the knot is the figure-eight knot, a positive $3$-braid knot, a negative $3$-braid knot or the connected sum of a $(2,r)$-torus knot and a $(2,s)$-torus knot for some odd integers $r,s \neq \pm 1$. In particular, we show that it is a $3$-braid knot. Taniyama and Yasuhara showed that the band-unknotting number of a knot is less than or equal to half the crossing number of the knot under our notation. We show that the equality holds if and only if the knot is the trivial knot or the figure-eight knot.