Computation of \tau invariant

I am trying to understand the following inequality, $$0 \leq \tau (K_{+}) - \tau(K_{-}) \leq 1$$ from the following paper by Livingston. \ https://arxiv.org/pdf/math/0311036.pdf . At page 737 , he argues, a negative crossing change converts $$-T_{2,3}$$ into the unknot.So $$(K_{+} \# -T_{2,3} \#-K_{-})$$ bounds a disks with 2 double point. Resolving the double point we get a genus 1 surface. I am trying to construct an explicit movie from $$K_{+}$$ to $$T_{2,3} \# K_{-}$$ with a genus one cobordism between them. I end up having a genus 2 cobordism instead of 1. Please help me with the movie move.