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Recently Witten introduced a type IIB brane construction with certainboundary conditions to study knot invariants and Khovanov homology. Theessential ingredients used in his work are the topologically twisted N = 4Yang-Mills theory, localization equations and surface operators. In this paperwe extend his construction in two possible ways. On one hand we show that aslight modification of Witten's brane construction could lead, using certainwell defined duality transformations, to the model used by Ooguri-Vafa to studyknot invariants using gravity duals. On the other hand, we argue that boththese constructions, of Witten and of Ooguri-Vafa, lead to two differentseven-dimensional manifolds in M-theory from where the topological theories mayappear from certain twisting of the G-flux action. The non-abelian nature ofthe topological action may also be studied if we take the wrapped M2-branestates in the theory. We discuss explicit constructions of theseven-dimensional manifolds in M-theory, and show that both the localizationequations and surface operators appear naturally from the Hamiltonian formalismof the theories. Knots and link invariants are then constructed using M2-branestates in both the models.

Knot Invariants and M-Theory I: Hitchin Equations, Chern-Simons Actions, and the Surface Operators