Recently I came across the following expository slides of Ciprian Manolescu: The unknotting problem: a journey from elementary to advanced mathematics, talk for the high school students at the Romanian Master of Mathematics competition, Bucharest, 2012
They are a fantastic and indeed high-school level introduction to Floer homology for knots. It is truly a marvel to eat this sweet mathematical fruit, borne of great toil by many great mathematicians, conjured from the Platonic beyond, clarified and simplified, to be finally presented with such nice diagrams for the viewing pleasure of all humankind!
However, I do not understand the last step of going from the "a graph (of a chain complex) to homology", pg. 23 of the slides:
Questions: What edges can we delete? When can we delete them? When can we delete vertices? Is that the only form of "zigzag" that can be replaced? I don't see a zigzag in the $3\times 3$ unknot example...