The Rasmussen invariant, four-genus and three-genus of an almost   positive knot are equal
  An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, $4$-genus and $3$-genus of a positive knot are equal. In this paper, we prove that the Rasmussen invariant, $4$-genus and $3$-genus of an almost positive knot are equal. Moreover, we determine the Rasmussen invariant of an almost positive knot in terms of its almost positive knot diagram. As corollaries, we prove that any almost positive knot is not homogeneous, and there is no almost positive knot of $4$-genus one. 
Keiji Tagami
math.GT