Rational homology spheres and the four-ball genus of knots

Abstract

Using the Heegaard Floer homology of Ozsváth and Szabó we investigate obstructions to definite intersection pairings bounded by rational homology spheres. As an application we obtain new lower bounds for the four-ball genus of Montesinos links.In this section we study the relationship between a smooth four-manifold X and its boundary Y . We will assume throughout that X is negative definite. The following is an extension of [3, Lemma 3].Lemma 2.1 Let Y be a rational homology sphere; denote by h the order of H 1 (Y ; Z). Suppose that Y bounds X and denote by s the absolute value of the determinant of the intersection pairing on H 2 (X, Z)/Tors. Then h = st 2 , where st is the order of the image of H 2 (X; Z) in H 2 (Y ; Z), and t is the order of the image of the torsion subgroup of H 2 (X; Z).Proof. Note that for b 2 (X) > 0, X has a non-degenerate integer intersection formwe denote the absolute value of the determinant of this pairing by s. If b 2 (X) = 0, then set s = 1. The long exact sequence of the pair (X, Y ) yields the following (with integer coefficients):

A characterisation of the $\mathbf{n\langle1\rangle \oplus\langle3\rangle}$ form and applications to rational homology spheres

“…Furthermore, if k(Y ) < 0 then Y bounds a positive-definite plumbing. For our conventions for lens spaces and Seifert fibred spaces see [8]. Recall in particular that (α i , β i ) are coprime pairs of integers with α i ≥ 2.…”

“…These theorems constrain the possible intersection forms that Y may bound. The above inequality is used in [8] to constrain intersection forms of a given rank bounded by Seifert fibred spaces, with application to four-ball genus of Montesinos links. In this paper we attempt to get constraints by finding a lower bound on the left-hand side of (1) which applies to forms of any rank.…”

Floer theory and low dimensional topology

“…All of them have the obvious functorial properties, behaving well for example under compositions of cobordisms. Another important property is that the image of the induced map These functorial properties are the basis for most applications and calculations; see for example Némethi [21] and Owens-Strle [23].…”

Heegaard Diagrams and Holomorphic Disks

“…This result is proved in [81]. The relationship with the four-ball genus is discussed in 4.4, where we discuss the concordance invariant of [82] and [88], and also the method of Owens and Strle [70]. Finally, in Subsection 4.5, we discuss an application to the problem of knots with unknotting number one from [74].…”

“…Further links between the ThurstonBennequin invariant and τ are explored by Plamenevskaya, see [86]. A different method for bounding g * (K) is given by Owens and Strle in [70], where they describe a method using the correction terms for the branched double-cover of S 3 along K, Σ(K). Under favorable circumstances, their method gives an obstruction for Murasugi's bound…”