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Title: Three-strand pretzel knots, knot Floer homology and concordance invariants
Author: Waite, Daniel
Awarding Body: University of Glasgow
Current Institution: University of Glasgow
Date of Award: 2020
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This thesis is concerned with determining the knot Floer homology and concordance invariants of pretzel knots, in particular three-strand 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 quasi-alternating, two-bridge and L-space knots - there are still many simple families for which knot Floer homology and the associated concordance invariants are not known. Recent work by Ozsvath-Szabo 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 well-known 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 Dai-Hom-Stroffregen-Truong. The benefit of such combinatorial constructions is that they do not rely on computation of the counts of pseudo-holomorphic representatives of Whitney disks in some high-dimensional space, unlike classical knot Floer homology. The grid homology developed by Manolescu-Ozsvath-Sarkar has the disadvantage that although one need not calculate these counts - since by construction all Whitney disks considered in this theory have a single pseudo-holomorphic 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 one-to-one 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 three-strand 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 well-understood grading information makes three-strand 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 three-strand pretzel knots of the form P(2a,-2b-1,2c+1) and P(2a,-2b-1,-2c-1). 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 slice-Bennequin inequality of Kawamura. In particular, the family of three-strand pretzel knots described by P(2a,-2b-1,-2c-1) 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 nu-invariants of the family P(2a,-2b-1,2c+1) using the sharper slice-Bennequin 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 Ozsvath-Szabo, it is first necessary in Chapter 4 to define the algebraic objects used in the construction: namely A-infinity-algebras, associated to every horizontal level of a knot diagram in the required form; DA-bimodules, associated to every Morse event such as crossings, maxima and minima; Type D structures, associated to upper knot diagrams; and A-infinity-modules, 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 three-strand pretzel knots admit knot diagrams in a certain form - see Figure 5.1 - a new A-infinity-module associated to the minima in these special knot diagrams will be defined in Section 4.6.2. This new A-infinity-module greatly simplifies the calculation of the invariant C(P(2a,-2b-1,2c+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 DA-bimodules defined by Ozsvath-Szabo in their algebraic construction, and introduced in Chapter 4, the Type D structure for upper knot diagrams of three-strand pretzel knots can be determined inductively. Under certain conditions, the tensor product between a DA-bimodule 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 DA-bimodules 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 three-strand 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 A-infinity-algebras explained in Chapter 4, and the diagrammatic representation of Type D structures: see for example Figure 5.3. For D a three-strand pretzel knot in the family P(2a,-2b-1,2c+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 A-infinity-module 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 Dai-Hom-Stroffregen-Truong in 2019. This is a linearly independent family of concordance invariants, extracted from what they call a reduced knot-like 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 three-strand pretzel knots P(2a,-2b-1,2c+1), this is not needed to compute the phi-invariants, 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 three-strand pretzel knots. In particular, the remaining examples of three-strand 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,-2b-1,2c+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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics