A survey on Heegaard Floer homology and concordance

Abstract

In this survey article, we discuss several different knot concordance invariants coming from the Heegaard Floer homology package of Ozsváth and Szabó. Along the way, we prove that if two knots are concordant, then their knot Floer complexes satisfy a certain type of stable equivalence.

Upsilon-type concordance invariants

“…(1) picture each Z 2 -generator U m · x of C on the planar lattice Z × Z ⊂ R 2 in position (A(x) − m, −m) ∈ Z × Z, (2) label each Z 2 -generator U m · x of C with its Maslov grading M (x) − 2m ∈ Z, (3) connect two Z 2 -generators U n · x and U m · y with a directed arrow if in the differential of U n · x the coefficient of U m · y is non-zero. In [16] Ozsváth and Szabó show how to associate to a knot K ⊂ S 3 a knot type complex CF K ∞ (K) whose filtered chain homotopy type only depends on the isotopy class of K. For a concise introduction to the background material see [6].…”

“…The quotient set CFK/ ∼ has a natural group structure: the sum is given by tensor product, the class of zero is the one represented by the Floer chain complex of the unknot CF K ∞ (U ), and the inverse of the class of a complex C is the one represented by its dual complex Hom(C, Z 2 [U, U −1 ]). Theorem 2.1 (Hom [6]). The map K → CF K ∞ (K) associating to a knot K ⊂ S 3 its knot Floer complex descends to a group homomorphism C → CFK/ ∼ .…”

Concordance to links with an unknotted component

“…We believe that this link is not concordant to any link in S 3 , but we are not able to prove this at the moment. We also make a remark that the above question is a natural generalization of a theorem of Adam Levine [Lev16] (see also [HLL18]), where he proved that there exists a knot in a homology sphere which is not smoothly concordant to any knot in S 3 . As far as the authors knowledge, it is not known if such a statement is true for the topological category.…”

Using secondary Upsilon invariants to rule out stable equivalence of knot complexes

“…Two Heegaard Floer knot complexes are called stably equivalent if an acyclic complex can be added to each complex to make them filtered chain homotopy equivalent. In [6], Hom showed that if two knots are concordant, then their knot complexes are stably equivalent. The concordance invariants τ , ε, Υ, Υ 2 are all invariants of the stable equivalence class (see [5,6,8,11,12]).…”

“…6), x 2 = (1, 2) ⊗ (3, 3), x 3 = (2, 1) ⊗(1,6), and x 4 = (2, 1) ⊗(3,3).CFK ∞ (T (2, 5)) 0 ⊗ CFK ∞ (T (5, 6)) ∞ (T (2, 5)) 1 ⊗ CFK ∞ (T (5, 6)) The value of s for which each element at grading 1 is on the line L 4 5 ,s .Taking the boundaries, we get:∂(x 1 ) = ((0, 2) + (1, 1)) ⊗ (1, 6) = (0, 2) ⊗ (1, 6) + (1, 1) ⊗ (1, 6) = A + (1, 1) ⊗ (1, 6), ∂(x 2 ) = ((0, 2) + (1, 1)) ⊗ (3, 3) = (0, 2) ⊗ (3, 3) + (1, 1) ⊗ (3, 3) = B + (1, 1) ⊗ (3, 3), ∂(x 3 ) = ((2, 0) + (1, 1)) ⊗ (1, 6) = (2, 0) ⊗ (1, 6) + (1, 1) ⊗ (1, 6), ∂(x 4 ) = ((2, 0) + (1, 1)) ⊗ (3, 3) = (2, 0) ⊗ (3, 3) + (1, 1) ⊗ (3, 3).Notice that if Equation 5.1 is to hold, it must be that b 1 = b 2 = 1. Since∂(x 1 + x 2 ) = A + B + (1, 1) ⊗ (1, 6) + (1, 1) ⊗ (3, 3),we need b 3 = b 4 = 1 in order to counteract the extra contributions of x 1 and x 2 .…”