Elementary filtration Given a simplicial complex K consisting of l simplices, there is a filtration such that 1.=KoCK1c…CKL=K 2. Ki and Ki+1 differs by a simplex oi+1i 3. The simplices are sorted according to their dimension
Elementary filtration Given a simplicial complex 𝐾 consisting of 𝑙 simplices, there is a filtration such that 1. ∅ = K0 ⊂ 𝐾1 ⊂ ⋯ ⊂ 𝐾𝑙 = 𝐾; 2. 𝐾𝑖 and 𝐾𝑖+1 differs by a simplex 𝜎𝑖+1 ; 3. The simplices are sorted according to their dimension
D1 K2
𝐾1 𝐾2 𝐾3 𝐾4 𝐾5 𝐾6 𝐾7 𝑣1 𝑣1 𝑣1 𝑣1 𝑣1 𝑣1 𝑣1 𝑣2 𝑣2 𝑣2 𝑣2 𝑣2 𝑣2 𝑣3 𝑣3 𝑣3 𝑣3 𝑣3 𝑒4 𝑒4 𝑒4 𝑒4 𝑒5 𝑒5 𝑒5 𝑒6 𝑒6 𝑡7
Elementary filtration Proposition Suppose Ki is an elementary filtration. Adding a p dimensional simplex Oj+1 to K; yields 1. Either1n(x+1)=B2(K)+1; 2.OrBp-1(K+1)=B-1(k)-1
Elementary filtration Proposition Suppose {𝐾𝑖 } is an elementary filtration. Adding a 𝑝 dimensional simplex 𝜎𝑗+1 to 𝐾𝑗 yields 1. Either 𝛽𝑝 𝐾𝑗+1 = 𝛽𝑝 𝐾𝑗 + 1; 2. Or 𝛽𝑝−1 𝐾𝑗+1 = 𝛽𝑝−1 𝐾𝑗 − 1
Pairing Adding o; causes situation(1): 0; is positive Adding o; causes situation(2): O is negative
Pairing • Adding 𝜎𝑗 causes situation (1): 𝜎𝑗 is positive; • Adding 𝜎𝑗 causes situation (2): 𝜎𝑗 is negative;
Pairing Pairing Theorem(Edelsbrunner/ Zomorodian) Given a simplicial complex k and its elementary filtration Ki, each negative simplex is paired with a unique positive simplex Remark The pairing is chosen so that the positive simplex is the youngest in the killed cycle If a positive simplex is not paired with any negative simplex, it's called paired with infinity
Pairing Pairing Theorem (Edelsbrunner/Zomorodian) Given a simplicial complex 𝐾 and its elementary filtration {𝐾𝑖 }, each negative simplex is paired with a unique positive simplex. Remark • The pairing is chosen so that the positive simplex is the youngest in the killed cycle. • If a positive simplex is not paired with any negative simplex, it’s called paired with infinity