Genera of some closed 4-braids

We obtain the genera of knots or links that are the closures of certain types of 4-braids. These 4-braids are written as positive words in the band-generator presentation of the 4-braid group considered by E.S. Kang et al. in this volume, and their closures naturally bound surfaces that consist of four disks and half-twisted bands connecting any two disks. In fact we show that these spanning surfaces have the minimal genera.