JournalsqtVol. 2, No. 3pp. 217–239

A note on sign conventions in link Floer homology

  • Sucharit Sarkar

    Columbia University, New York
A note on sign conventions in link Floer homology cover
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Abstract

For knots in S3S^3, the bi-graded hat version of knot Floer homology is defined over Z\mathbb{Z}; however, for an ll-component link LL in S3S^3 with l>1l>1, there are 2l12^{l-1} bi-graded hat versions of link Floer homology defined over Z\mathbb{Z}; the multi-graded hat version of link Floer homology, defined from holomorphic considerations, is only defined over F2\mathbb{F}_2; and there is a multi-graded version of link Floer homology defined over Z\mathbb{Z} using grid diagrams. In this short note, we try to address this issue, by extending the F2\mathbb{F}_2-valued multi-graded link Floer homology theory to 2l12^{l-1} Z\mathbb{Z}-valued theories. A grid diagram representing a link gives rise to a chain complex over F2\mathbb{F}_2, whose homology is related to the multi-graded hat version of link Floer homology of that link over F2\mathbb{F}_2. A sign refinement of the chain complex exists, and for knots, we establish that the sign refinement does indeed correspond to the sign assignment for the hat version of the knot Floer homology. For links, we create 2l12^{l-1} sign assignments on the grid diagrams, and show that they are related to the 2l12^{l-1} multi-graded hat versions of link Floer homology over Z\mathbb{Z}, and one of them corresponds to the existing sign refinement of the grid chain complex.

Cite this article

Sucharit Sarkar, A note on sign conventions in link Floer homology. Quantum Topol. 2 (2011), no. 3, pp. 217–239

DOI 10.4171/QT/20