Knot doubling operators and bordered Heegaard Floer homology (Q2914384)

This paper is one of the first applications of the recently defined algebraic tool for the computation of Heegaard Floer homology. The author uses bordered Floer homology [\textit{R. Lipshitz, P. S. Ozsváth} and \textit{D. P. Thurston}, Quantum Topol. 2, No. 4, 381--449 (2011; Zbl 1256.57012)] to give obstructions for smooth sliceness for some generalizations of Whitehead doubles of knots. In particular the author proves that the \(\tau\) invariant [\textit{P. S. Ozsváth} and \textit{Z. Szabó}, Geom. Topol. 7, 615--639 (2003; Zbl 1037.57027)] of a family of satellite knots obtained via twisted infection along two components of the Borromean rings depends only on the two twisting parameters and the values of the \(\tau\) invariants for the two companion knots.NEWLINENEWLINEThe idea of the proof is that this twisted infection can be described as one unknot component of the Borromean rings in \(S^3\), which is given as a union of the complement of the other two components of the Borromean rings and the complements of the companion knots. Thus the knot Floer homology for the twisted infection can be computed as the tensor product of the bordered Floer homology for one unknot component of the Borromean rings in the complement of the other two components, and the bordered Floer homologies for the complements of the companion knots.NEWLINENEWLINEThe bordered Floer homology of knot complements can be computed from the knot Floer homologies. Then the author gives an explicit bordered Heegard diagram for one unknot component of the Borromean rings in the complement of the other two components, and computes its bordered Heegaard Floer homology. The most involved (and thus most technical) part of the proof is the understanding of the tensor product. The author manages to understand enough properties of the tensor product, and thus the knot Floer homology of the twisted infection, from the partial data (of the value of the \(\tau\) invariant) of the knot Floer homology of the companion knots to determine the \(\tau\) invariant of the twisted infection.