This Ph.D. dissertation studies the operation of Murasugi sum on pairs of knots (or links) inthe three-sphere S3. This operation, which produces a knot (or link) involves several choices which usually change the isotopy class of the produced object. This could lead one to believe the Murasugi sum has no hope of preserving properties of knots (or links). Indeed, without restricting the involved choices, we show in Chapter 2 that any knot is the Murasugi sum of any two knots. However, we also show that restricting the possible choices restricts the possible Murasugi sums. The contents of Chapter 2 are joint with Mikami Hirasawa. Historically, the usual restriction has been to consider knots which are Murasugi summed along minimal genus Seifert surfaces. In this setting, knot genus is additive under Murasugi sums, and the rank of the "top" group of knot Floer homology HFK is multiplicative under Murasugi sums. Continuing this trend in Chapter 4, we study the symplectic Floer homology HF∗ of a family of knots which are closures of a particular type of 3-string braid. These knots can be viewed as Murasugi sums performed along minimal genus Seifert surfaces, and we show that a large range of choices in these Murasugi sums all yield the same rank of HF∗. We carry out these calculations via the Bestvina-Handel algorithm and the combinatorial formula for HF∗ of pseudo-Anosov maps due to Cotton-Clay. We hope that these calculations shed some light on the behavior of HF∗ under Murasugi sums, as this group has garnered recent interest for its connection to the "next-to-top" group of knot Floer homology.
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