A spanning tree cohomology theory for links
There are many new knot invariants including the Khovanov homology, which is a sequence of homology groups whose Euler characteristic is the Jones polynomial, the knot Floer homology of Ozsváth and Szabó, which is similarly related to the Alexander polynomial, and the Heegaard-Floer homology of the branched double cover along a link. The Heegaard-Floer homology was introduced by \textit{P. Ozsváth} and \textit{Z. Szabó} [Duke Math. J. 129, No. 1, 39--61 (2005; Zbl 1083.57042)] as a link invariant of the branched double cover \(\Sigma(L)\) of \(S^3\) along a link \(L\), and they discovered a spectral sequence from Khovanov homology to \(\widehat{HF}(\Sigma(L))\) such that every \(E_r\)-term of this sequence is a link invariant. Recently, \textit{J. Baldwin}, \textit{P. Ozsváth}, and \textit{Z. Szabó} [Heegaard Floer homology of double-covers, Kauffman states, and Novikov rings, to appear] have introduced a perturbed Heegaard-Floer homology with coefficients in a Novikov ring, constructed an appropriate BOS-spectral sequence, and showed that the \(E_2\)-term is trivial and the cochain complex \((E_2,d_2)\) has a combinatorial description. The main purpose of this paper is to show that the \(E_3\)-term of the BOS-spectral sequence is an invariant of oriented links. Also, the authors present concrete examples of computation of this invariant. If \(L\) is an oriented link in \(S^3\) with generic projection \({\mathcal D}\) such that every connected component of \(S^3\smallsetminus{\mathcal D}\) is simply connected, then \((C({\mathcal D}),\Phi)\) is the cochain complex which is the \(E_2\)-term of the BOS-spectral sequence converging to the Heegaard-Floer homology \(\widehat{HF}(\sum(L))\), where \(\Phi(T)=\sum\limits_{T'}\left(\frac{1}{1+\alpha(T,T')}+\frac{1}{1+\beta(T,T')}\right)T'\) for a spanning tree \(T\). They prove that if \(F\) is a field of characteristic \(2\) with elements \(u_f\), \(z_v\), then for each \(i\), \((*)\,\text{rank}_F\left(H^i(C({\mathcal D},F,(u_f),(z_v)),\Phi)\right)\) is independent of the choice of \(F\), and of the projection \({\mathcal D}\) of an oriented link \(L\). Moreover, if \(L\) is a knot, then \(\Phi(T)\) is independent of orientation, and if \(L\) is a link which has a projection with more than \(1\) connected component, then \((*)=0\).
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