Abstract: Over the past two decades, knot Floer homology has emerged as a powerful tool in knot theory, offering a rich array of applications. As a categorification of the Alexander polynomial, it not only detects the unknot, the Seifert genus, and the fiberedness of knots but also provides strong lower bounds for invariants such as the 4-ball genus, the unknotting number, and the bridge number.
The skein relation for the Alexander polynomial finds its deeper counterpart in the skein exact triangle for knot Floer homology, a versatile tool for both computation and theoretical exploration. Additionally, Manolescu's construction of an unoriented skein exact triangle establishes a fascinating connection to Khovanov homology. This raises a natural question: Are there other exact triangles in knot Floer homology?
In this talk, we begin with a brief overview of knot Floer homology and its key applications. We then delve into the construction of skein exact triangles and present several new examples, expanding the scope of this powerful framework.
