This thesis is concerned with determining the knot Floer homology and concordance invariants of pretzel knots, in particular threestrand pretzel knots. Knot Floer homology is a package of knot invariants developed by Ozsvath and Szabo, and despite the invariants being known for simple classes of knots  for example quasialternating, twobridge and Lspace knots  there are still many simple families for which knot Floer homology and the associated concordance invariants are not known. Recent work by OzsvathSzabo developed a construction of an algebraic invariant C(D), conjectured by them to be equal to a variant of knot Floer homology. This complex is a bigraded, bifiltered chain complex whose filtered chain homotopy type is an invariant of a knot. Their construction  which has also been implemented in a C++ program  is a divide and conquer method which decomposes knot diagrams in a certain form into smaller pieces, to which algebraic objects are then associated. These algebraic objects are themselves invariants (up to appropriate equivalence) of partial knot diagrams, and are pieced together to form the full invariant. As with classical knot Floer homology, one can study the homology of this complex C(D), or the homology of subcomplexes and quotient complexes, which are also invariants of a knot. Even more recent work of Ozsvath and Szabo confirms that this conjectured equivalence between the theories holds. Hence, like the wellknown grid homology of a knot, this algebraic method provides a combinatorial construction of knot Floer homology  or in this case some slightly modified version of classical knot Floer homology, like that presented by DaiHomStroffregenTruong. The benefit of such combinatorial constructions is that they do not rely on computation of the counts of pseudoholomorphic representatives of Whitney disks in some highdimensional space, unlike classical knot Floer homology. The grid homology developed by ManolescuOzsvathSarkar has the disadvantage that although one need not calculate these counts  since by construction all Whitney disks considered in this theory have a single pseudoholomorphic representative  this is at the expense of computing the homology of chain complexes with a very large number of generators (relative to crossing number). However, the algebraic invariant C(D) of Ozsvath and Szabo has the form of a chain complex whose generators are in onetoone correspondence with the Kauffman states of a knot diagram. Kauffman states are decorated, oriented knot projections, and the bigrading of the corresponding generators can be determined from the Kauffman states. Similarly, classical knot Floer homology can also be calculated from a chain complex generated by Kauffman states. Adapting the work of Eftekhary, the Kauffman states for a threestrand pretzel knot P can be placed into three families, based upon the positions of the decorations on each of the three strands. These families have grading information that is determined by the positions of the decorations on each strand  see Table 2.1 and Table 2.2 for explicit calculations of these gradings. Using the grading information associated to these Kauffman states, one can restrict the possible differentials within the full knot Floer chain complex of P, as demonstrated by Lemma 2.10. Furthermore, the classification of the Kauffman states into these three families with wellunderstood grading information makes threestrand pretzel knots particularly amenable to study using the divide and conquer construction of Ozsvath and Szabo. After an introduction to knot Floer homology and the current knowledge for pretzel knots and links provided in Chapter 1, this thesis will present in Chapter 2 a definition of Kauffman states, their grading information, and in particular the possible Kauffman states for threestrand pretzel knots of the form P(2a,2b1,2c+1) and P(2a,2b1,2c1). Moreover, in Chapter 2, it will be demonstrated how the grading information of the Kauffman states for these pretzel knots can be used to restrict the possible Maslov disks between generators of the classical knot Floer homology. In so doing, one can read off certain knot Floer homology groups directly from the combinatorial information, see for example Lemma 2.7 and Lemma 2.9. Chapter 3 defines many of the simpler concordance invariants extracted from classical knot Floer homology, and in particular Section 3.3 describes how the concordance invariants of some families of pretzel knots can be bounded by using the sharper sliceBennequin inequality of Kawamura. In particular, the family of threestrand pretzel knots described by P(2a,2b1,2c1) for a,b and c natural numbers are quasipositive, and so have concordance invariants nu and tau equal to their Seifert genus. Furthermore, one can place bounds upon the tau and nuinvariants of the family P(2a,2b1,2c+1) using the sharper sliceBennequin inequality and work of Kawamura, and what is more, these bounds are strong enough to determine these concordance invariants the case of b >= c, as demonstrated by Lemma 3.1*. Before describing the construction of the algebraic invariant C(D) defined by OzsvathSzabo, it is first necessary in Chapter 4 to define the algebraic objects used in the construction: namely Ainfinityalgebras, associated to every horizontal level of a knot diagram in the required form; DAbimodules, associated to every Morse event such as crossings, maxima and minima; Type D structures, associated to upper knot diagrams; and Ainfinitymodules, associated to lower knot diagrams. In this chapter, the specific algebraic objects used in the construction of C(D) are defined over the required differential graded algebras. Furthermore, because all threestrand pretzel knots admit knot diagrams in a certain form  see Figure 5.1  a new Ainfinitymodule associated to the minima in these special knot diagrams will be defined in Section 4.6.2. This new Ainfinitymodule greatly simplifies the calculation of the invariant C(P(2a,2b1,2c+1)), allowing the inductive proofs presented in Chapter 5 determining this invariant to be more closely motivated by the Heegaard diagrams for this family of knots used by Eftekhary. Using the DAbimodules defined by OzsvathSzabo in their algebraic construction, and introduced in Chapter 4, the Type D structure for upper knot diagrams of threestrand pretzel knots can be determined inductively. Under certain conditions, the tensor product between a DAbimodule and a Type D structure can be taken to yield another Type D structure. This process is outlined in Section 4.5. Intuitively, since Type D structures are associated to upper knot diagrams, and DAbimodules to Morse events (such as crossings or maxima), attaching a Morse event to an upper knot diagram yields another upper knot diagram. The generators of Type D structures are in bijection with upper Kauffman states, and for threestrand pretzel knots the upper Kauffman states can also be separated into distinct families based upon the decorations on each strand. This separation of upper Kauffman states into families allows one to determine the Type D structure after an arbitrary number of crossings in each strand. In the proofs in Chapter 5, much use is made of both the truncation of the Ainfinityalgebras explained in Chapter 4, and the diagrammatic representation of Type D structures: see for example Figure 5.3. For D a threestrand pretzel knot in the family P(2a,2b1,2c+1), the structure of C(D)  and the associated homology theories recently proven to be equivalent to the hat and minus version of knot Floer homology  will be determined in Chapter 6, relying on the inductive computations of Chapter 5 and the construction of a new Ainfinitymodule associated to the minima of a special knot diagram for these knots outlined in Section 4.6.2. From these homology theories, the invariants nu and tau will be determined. These invariants were defined by Ozsvath and Szabo, and although they are now proven to be equivalent to the familiar concordance invariants in knot Floer homology, they are themselves invariants of the local equivalence class of the bigraded complex C(D). In Section 6.2.3 these invariants are demonstrated to also be additive under connected sum. This is as a corollary of the fact that the complex C(D # E) satisfies the Kunneth relation, see Proposition 6.16. Theorem 6.6, determining the homology theory H(C^{}(D)), is also sufficient to determine the infinite family of concordance invariants phi, introduced by DaiHomStroffregenTruong in 2019. This is a linearly independent family of concordance invariants, extracted from what they call a reduced knotlike complex. Since the complex C(D) is equivalent to the complex defined by them, one could also simplify C(D) to a reduced knot like complex. However, in the case of the threestrand pretzel knots P(2a,2b1,2c+1), this is not needed to compute the phiinvariants, as demonstrated by Lemma 6.14 Chapter 6 finishes by suggesting new areas where the techniques outlined within this thesis might be employed, and open problems in the study of threestrand pretzel knots. In particular, the remaining examples of threestrand pretzel knots whose slice genus is not known will be discussed. The concordance invariants defined in Chapter 6 are insufficient to answer these open questions; it is hoped, however, that since C(D) provides more information than the minus and hat versions of knot Floer homology, Theorem 6.1 determining C(P(2a,2b1,2c+1)) might prove useful for answering these questions in the future. Figures within this thesis have been constructed by the author using the vector drawing package IPE. Where these have been adapted from existing figures in other works, this has been appropriately cited.
