An intutive proof of 'replacing two-caps by a handle'

Question

I am trying to understand a statement given in Polchinski Vol.1 - a torus with cross-cap can be obtained either as (g,b,c) = (0,0,3) or as (1,0,1), trading two cross-caps for a handle.

Here, g is the genus, b is no. of boundaries and c is the no. of cross-caps. I am looking for an intuitive proof of the statement, without involving a lot of mathematics.

Answer 1

Read Conway's zip proof at Conway.