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Heegaard Floer homology
of spatial graphs
Shelly Harvey and Danielle O’Donnol |
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Algebraic & Geometric Topology 17 (2017) 1445–1525
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Abstract |
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We extend the theory of combinatorial link Floer homology to a class of oriented spatial graphs called transverse spatial graphs. To do this, we define the notion of a grid diagram representing a transverse spatial graph, which we call a graph grid diagram. We prove that two graph grid diagrams representing the same transverse spatial graph are related by a sequence of graph grid moves, generalizing the work of Cromwell for links. For a graph grid diagram representing a transverse spatial graph , we define a relatively bigraded chain complex (which is a module over a multivariable polynomial ring) and show that its homology is preserved under the graph grid moves; hence it is an invariant of the transverse spatial graph. In fact, we define both a minus and hat version. Taking the graded Euler characteristic of the homology of the hat version gives an Alexander type polynomial for the transverse spatial graph. Specifically, for each transverse spatial graph , we define a balanced sutured manifold . We show that the graded Euler characteristic is the same as the torsion of defined by S Friedl, A Juhász, and J Rasmussen. |
Keywords
spatial graphs, Heegaard Floer homology, grid diagrams,
embedded graph
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Mathematical Subject Classification 2010
Primary: 57M15
Secondary: 05C10
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References |
Publication
Received: 23 September 2015
Revised: 11 October 2016
Accepted: 28 October 2016
Published: 17 July 2017
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Authors |
| Shelly Harvey | |
| Department of Mathematics Rice University MS 136, PO Box 1892 Houston, TX 77251-1892 United States |
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| Danielle O’Donnol | |
| Department of Mathematics Indiana University Rawles Hall 831 E 3rd St Bloomington, IN 47505 United States |
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