On the impossibility of a generalization of the HOMFLY-polynomial to labelled oriented graphs
Let \(D=\langle x_1,x_2,...,x_n|R_1,R_2,...,R_n\rangle\) be a group presentation, where each relator is of the form \(x_ix_j=x_jx_k\). We can assign a labelled oriented graph \(\Gamma(D)\) to \(D\) by defining a vertex \(i\) for every generator \(x_i\) and an oriented edge from \(i\) to \(k\) labelled by \(j\) for the relator \(x_ix_j=x_jx_k\). For a regular diagram of an oriented link \(L\) in \({\mathbb R}^3\), a Wirtinger presentation \(D\) of a group \(\pi_1({\mathbb R}^3\smallsetminus L)\) has the form of a labelled oriented graph \(\Gamma(D)\) with each component a circle or a single vertex. Let \(\mathcal C\) be the set of the labelled oriented graphs with each component a circle or a single vertex. We note that a local change of a regular diagram of a link \(L\) induces a local change of a Wirtinger presentation \(D\) of \(\pi_1({\mathbb R}^3\smallsetminus L)\), and also a local change of \(\Gamma(D)\). This implies Reidemeister moves and HOMFLY polynomial have algebraic analogues for \(\mathcal C\). The author shows that an algebraic generalization of HOMFLY polynomial to \(\mathcal C\) is almost trivial. More precisely if HOMFLY polynomial of a labelled oriented graph \(\Gamma\) in \(\mathcal C\) is an invariant under Reidemeister moves, then it only detects the parity of the number of components of \(\Gamma\).
scientific article; zbMATH DE number 1013900
scientific article