The pillowcase and traceless representations of knot groups II: a Lagrangian–Floer theory in the pillowcase

Abstract

We define an elementary relatively Z/4 graded Lagrangian-Floer chain complex for restricted immersions of compact 1-manifolds into the pillowcase, and apply it to the intersection diagram obtained by taking traceless SU (2) character varieties of 2-tangle decompositions of knots. Calculations for torus knots are explained in terms of pictures in the punctured plane. The relation to the reduced instanton homology of knots is explored.

The Fukaya category of the pillowcase, traceless character varieties, and Khovanov cohomology

“…This article continues the investigation, set in motion by our earlier articles [14, 13,18], of the information contained in the traceless SU (2) character varieties associated to a 2-stranded tangle decomposition of the link. In [13] we considered a tangle decomposition (S 3 , L) = (D 3 , T 0 ) ∪ (S 2 ,4) (D 3 , T 1 ), where T 0 is a trivial 2-tangle in a 3-ball and T 1 its complement, and defined a Lagrangian Floer complex for L from two variants of the immersed traceless character varieties of the complements of the tangles in the decomposition, R π (D 3 , T 0 ), R π (D 3 , T 1 ). This theory, which we called pillowcase homology, takes place in the smooth part P * of the traceless character variety of the 4-punctured Conway sphere, P = R(S 2 , 4), a variety easily identified with the the quotient of the torus by the hyperelliptic involution.…”

“…Such considerations lead us to take specific choices of gradings on L 1 , W 0 , and W 1 , and are made precise in the following lemma. 1 The line field λ coincides with the line field called λinst in [13], which arises as a consequence of the Z/4-grading on Kronheimer-Mrowka's singular instanton homology [23] and a splitting theorem for spectral flow. For any λ grading of L 0 , there are uniquely determined λ gradings of L 1 , W 0 and W 1 such that…”

“…We place this base point at 1 ∈ B 2n ⊂ ∂D 2 and call it the earring. The terminology comes from [14,13] (see also Appendix B) and corresponds to the gauge-theoretic construction introduced in [23]. It also serves to distinguish it from the dots introduced below in Bar-Natan's dotted cobordism category.…”

“…We arrived at the constructions of the present paper by attempting to iterate an exact triangle in the Fukaya category of the pillowcase arising from the traceless character varieties of the three 2-stranded tangles defining the unoriented skein relation. The pillowcase homology of [13] mentioned above possesses a long exact sequence for the unoriented skein relation as a consequence of this exact triangle in the Fukaya category. The present work was an attempt to bypass the computational difficulties involved in (i) understanding the Lagrangians associated to traceless character varieties of tangles, and (ii) proving that the pillowcase homology associated to two such Lagrangians is an invariant of the underlying link.…”

“…We relegate the rather technical work involved in these calculations to Appendix A; there, we indicate how the calculations are carried out, and in particular why the relevant µ n vanish for n ≥ 4. Section 5 discusses gradings and we construct Z-gradings in L (lifting relative Z/4gradings coming from the gauge theoretic considerations of [13]). Section 6 introduces a variant of Bar-Natan's [8] 2-tangle category from which we produce, in Section 6, an A ∞ -functor to our Fukaya category L. This functor extends to a functor on the triangulated envelope of twisted complexes, and the image of Bar-Natan's twisted complex associated to a tangle is the twisted complex over L which we assign to a tangle diagram.…”