The difference between the boundary maps of 2 tetrahedrons

Question

Here is the first figure on pg.105 in Allen Hatcher:

But I am not sure why the negative sign before $[v_0, v_2, v_3]$ and $[v_0, v_1, v_2]$ in the boundary map. Could someone explain this to me please?

Also, how the boundary map of the previous figure differ from the boundary map of the following figure(on pg.104 of Allen Hatcher), could someone explain this to me please?

As far as I understand, the previous figure has 4 zero simplices, 6 one simplices and 3 two simplices, am I correct?

Answer 1

It's related to the parity of a permuation.

Changing a vertex and changing the order of vertices by an adjacent transposition both change the sign.

The definition of the boundary operator for simplices is given here:

https://en.wikipedia.org/wiki/Simplicial_homology#Boundaries_and_cycles