Knot invariants arising from homological operations on Khovanov homology (Q2799086)

This paper will be of use for anyone interested in knot invariants arising from Khovanov homology. One of the main results of the paper is the construction of an algebra which is generated by two Bockstein operations for homology with coefficients in \(\mathbb Z_2\): \(\beta_e:\mathcal H^i_{\mathbb Z_2}(L)\longrightarrow\mathcal H^{i+1}_{\mathbb Z_2}(L)\), \(\beta_o:\mathcal H^i_{\mathbb Z_2}(L)\longrightarrow\mathcal H^{i+1}_{\mathbb Z_2}(L)\).NEWLINENEWLINEThe authors lift this algebra to integral Khovanov homology and prove these two algebras are infinite.NEWLINENEWLINEThe integral lift: \(\varphi_{oe}:\mathcal H^i_o(L)\longrightarrow\mathcal H^{i+1}_e(L)\), \(\varphi_{eo}:\mathcal H^i_e(L)\longrightarrow\mathcal H^{i+1}_o(L)\).NEWLINENEWLINEIn [Banach Cent. Publ. 103, 291--355 (2014; Zbl 1336.57024)], the first author proposed that the unified Khovanov homology \(\mathcal H_\xi(L)\) over \(\mathbb Z_\xi\) can be regarded as an extension of both \(\mathcal H_e(L)\) and \(\mathcal H_o(L)\), but it is not immediately clear whether \(\mathcal H_\xi(L)\) is a stronger invariant than \(\mathcal H_e(L)\) and \(\mathcal H_o(L)\). It is shown in the paper that \(\mathcal H_\xi(L)\) is a finer link invariant than \(\mathcal H_e(L)\oplus \mathcal H_o(L)\) through a list of examples.NEWLINENEWLINEThe case of reduced Khovanov homology is also considered.