Grid triples, link cobordisms in CPbar and the computation of Heegaard-Floer TQFT
Abstract
We give a combinatorial description of the Heegaard--Floer link-cobordism maps associated with pointed triple grid diagrams in the complement of three $4$-balls in complex projective plane with reverse orientation. Every contributing index-zero holomorphic triangle decomposes into Euclidean triangular sheets. We characterize the compatible collections of sheets that contribute by an explicit geometric separation criterion, expressed as a lattice-point test, and prove that each has mod-2 count one. Individual sheets may wrap around the torus in the unblocked theory, whereas they are embedded in the blocked theory. In each fixed SpinC structure, the map is determined by a finite enumeration. A difference number computed from the triangular regions determines the SpinC structure of a Whitney triangle. Together with the topology of the associated surface, this gives an explicit formula for the absolute bigrading of each summand of the cobordism map. After basepoint reduction, the formula determines the Laurent monomial representing any nonzero map on the rank-one torsion-free two-variable knot Floer modules, yielding vanishing and ideal-containment criteria. We apply the construction to an affine family of grid links K(n;p,s), obtaining explicit cobordisms involving infinite families of torus knots and links, together with closed formulas for the bigradings of their distinguished grid states.
Related
BibTeX
@misc{mahkam2026grid,
title = {{Grid triples, link cobordisms in $\CPbar$ and the computation of Heegaard-Floer TQFT}},
author = {Roohallah Mahkam},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03714},
eprint = {2611.03714},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03714},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Grid triples, link cobordisms in CPbar and the computation of Heegaard-Floer TQFT
AU - Roohallah Mahkam
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.03714
UR - https://arxiv.org/abs/2611.03714
AB - We give a combinatorial description of the Heegaard--Floer link-cobordism maps associated with pointed triple grid diagrams in the complement of three $4$-balls in complex projective plane with reverse orientation. Every contributing index-zero holomorphic triangle decomposes into Euclidean triangular sheets. We characterize the compatible collections of sheets that contribute by an explicit geometric separation criterion, expressed as a lattice-point test, and prove that each has mod-2 count one. Individual sheets may wrap around the torus in the unblocked theory, whereas they are embedded in the blocked theory. In each fixed SpinC structure, the map is determined by a finite enumeration. A difference number computed from the triangular regions determines the SpinC structure of a Whitney triangle. Together with the topology of the associated surface, this gives an explicit formula for the absolute bigrading of each summand of the cobordism map. After basepoint reduction, the formula determines the Laurent monomial representing any nonzero map on the rank-one torsion-free two-variable knot Floer modules, yielding vanishing and ideal-containment criteria. We apply the construction to an affine family of grid links K(n;p,s), obtaining explicit cobordisms involving infinite families of torus knots and links, together with closed formulas for the bigradings of their distinguished grid states.
ER -