Semigroups, \(d\)-invariants and deformations of cuspidal singular points of plane curves (Q2801741)

This paper studies \(\delta\)-constant deformations of irreducible plane curve singularities. The main result is a topological proof for the semicontinuity of semigroups: suppose that a curve singularity with semigroup \(S_0\) deforms into \(n\) irreducible singularities with semigroups \(S_1,\dots, S_n\), then for any integer \(m\geq0\) and nonnegative \(m_i\) with \(m_1+\cdots m_n=m\) one has NEWLINE\[NEWLINE \# S_0\cap[0,m)\leq \sum_{j=1}^n \#S_j\cap [0,m_j)\;. NEWLINE\]NEWLINE The proof is based on an inequality for \(d\)-invariants from Heegard-Floer theory. Given a \(\delta\)-constant deformation of an irreducible curve singularities into irreducible curve singularities, there exists a \textit{positively self-intersecting concordance} from the link of the central fibre to the connected sum of the links of deformed curves; by definition, a positively self-intersecting concordance from an oriented knot \(K_0\) to an oriented knot \(K_1\) is a smoothly immersed annulus \(A\) in \(S^3\times [0,1]\) with \(\partial(S^3\times [0,1],A) =-(s^3,K_0)\cup(S^3,K_1)\) whose image has only positive transverse self-intersections. Note that this relation is not symmetric. From such a concordance a cobordism \(W_q\) between \(S^3_q(K_0)\) and \(S^3_q(K_1)\) is constructed, to which Heegard-Floer theory applies. As application it is shown that if a torus knot \(T(a,b)\) appears in some minimal unknotting sequence for \(T(c,d)\), then \(a\leq c\). The authors also give an example where semicontinuity of the semigroup excludes a deformation, while semicontinuity of the spectrum doesn't, although in general the spectrum is a more powerful invariant. In the last section the results are generalised to arbitrary knots. Heegard-Floer theory allows to define a gap sequence (which in general is no longer the complement of a semigroup), and the main theorem generalises to this situation: a positively self-intersecting concordance implies a inequality for gap functions, which in the algebraic case translates in the inequalities above.