In this talk, I’ll introduce a special family of (1,1) knots called constrained knots, which generalizes 2-bridge knots in the 3-sphere. I’ll provide a complete classification of constrained knots based on 5 parameters. Dimensions of Knot Floer homology and instanton knot homology for constrained knots are equal, which provides more examples of a conjecture made by Kronheimer and Mrowka. I’ll also discuss the relation between constrained knots and hyperbolic 1-cusped manifolds. In particular, I’ll show many hyperbolic 1-cusped manifolds with simple ideal triangulations are complements of constrained knots. Some results are joint work with John A. Baldwin and Zhenkun Li.

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