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 choiceswhich usually change the isotopy class of the produced object. This could lead one to believethe Murasugi sum has no hope of preserving properties of knots (or links). Indeed, withoutrestricting the involved choices, we show in Chapter 2 that any knot is the Murasugi sumof any two knots. However, we also show that restricting the possible choices restricts thepossible Murasugi sums. The contents of Chapter 2 are joint with Mikami Hirasawa.Historically, the usual restriction has been to consider knots which are Murasugi summedalong minimal genus Seifert surfaces. In this setting, knot genus is additive under Murasugisums, and the rank of the "top" group of knot Floer homology HFK is multiplicativeunder Murasugi sums. Continuing this trend in Chapter 4, we study the symplecticFloer homology HF⁸́₇ of a family of knots which are closures of a particular type of 3-stringbraid. These knots can be viewed as Murasugi sums performed along minimal genus Seifertsurfaces, and we show that a large range of choices in these Murasugi sums all yield thesame rank of HF⁸́₇. We carry out these calculations via the Bestvina-Handel algorithmand the combinatorial formula for HF⁸́₇ of pseudo-Anosov maps due to Cotton-Clay. Wehope that these calculations shed some light on the behavior of HF⁸́₇ under Murasugi sums,as this group has garnered recent interest for its connection to the "next-to-top" group ofknot Floer homology.
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