We describe a Dehn type surgery along a Legendrian-transverse knot K in a bi-contact structure. We show that, when the bi-contact structure defines an Anosov flow, there is a strong connection between the Anosovity of the new flow and contact geometry. We give an application to the geodesic flow on the unit tangent bundle of an hyperbolic surface. In contrast to the existing Dehn type surgeries on contact Anosov flows, for a vast class of knots our procedure does not require any restriction on the slope of the twist to generate new contact Anosov flows. We finally show that there are connections between our construction and the ones defined by Handel-Thurston, Fried-Goodman and Foulon-Hasselblatt.

We introduce a generalization of Goodman surgery to the category of projectively Anosov flows. This construction is performed along a knot that is simultaneously Legendrian and transverse for a supporting bi-contact structure. When the flow is Anosov, our operation generates the same flows of Goodman's construction. The Anosovity of the new flow is strictly connected to contact geometry. We use this relation to give a refinement to the sequence of surgeries coefficients producing Anosov flows in concrete examples. We give an interpretation of the bi-contact surgery in terms of contact-Legendrian surgery and admissible-inadmissible transverse surgery and we deduce some (hyper)tightness result for contact and transverse surgeries. Outside of the realm of Anosov flows, we produce new projectively Anosov flows on hyperbolic 3-manifolds with leaves of genus g > 1 in the invariant foliations.

We show that the contact gluing map of Honda, Kazez, and Matic has a natural algebraic description. In particular, we establish a conjecture of Zarev, that his gluing map on sutured Floer homology is equivalent to the contact gluing map.1 Homology groups are over F2-coefficients through the entirety of this paper.

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