Browsing by Author "Harvey, Shelly"
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Item A new filtration of the Magnus kernel(2013-09-16) McNeill, Reagin; Harvey, Shelly; Cochran, Tim D.; Riviere, Beatrice M.For a oriented genus g surface with one boundary component, S_g, the Torelli group is the group of orientation preserving homeomorphisms of S_g that induce the identity on homology. The Magnus representation of the Torelli group represents the action on F/F'' where F=π_1(S_g) and F'' is the second term of the derived series. I show that the kernel of the Magnus representation, Mag(S_g), is highly non-trivial and has a rich structure as a group. Specifically, I define an infinite filtration of Mag(S_g) by subgroups, called the higher order Magnus subgroups, M_k(S_g). I develop methods for generating nontrivial mapping classes in M_k(S_g) for all k and g≥2. I show that for each k the quotient M_k(S_g)/M_{k+1}(S_g) contains a subgroup isomorphic to a lower central series quotient of free groups E(g-1)_k/E(g-1)_{k+1}. Finally I show that for g≥3 the quotient M_k(S_g)/M_{k+1}(S_g) surjects onto an infinite rank torsion free abelian group. To do this, I define a Johnson-type homomorphism on each higher order Magnus subgroup quotient and show it has a highly non-trivial image.Item Casson towers and filtrations of the smooth knot concordance group(2014-04-16) Ray, Arunima; Cochran, Tim D.; Harvey, Shelly; Warren, JoeThe 4-dimensional equivalence relation of concordance (smooth or topological) gives a group structure on the set of knots, under the connected-sum operation. The n-solvable filtration of the knot concordance group (denoted C), due to Cochran-Orr-Teichner, has been instrumental in the study of knot concordance in recent years. Part of its significance is due to the fact that certain geometric attributes of a knot imply membership in various levels of the filtration. We show the counterpart of this fact for two new filtrations of C due to Cochran-Harvey-Horn, the positive and negative filtrations. The positive and negative filtrations have definitions similar to that of the n-solvable filtration, but have the ability (unlike the n-solvable filtrations) to distinguish between smooth and topological concordance. Our geometric counterparts for the positive and negative filtrations of C are defined in terms of Casson towers, 4-dimensional objects which approximate disks in a precise manner. We establish several relationships between these new Casson tower filtrations and the various previously known filtrations of C, such as the n-solvable, positive, negative, and grope filtrations. These relationships allow us to draw connections between some well-known open questions in the field.Item Derivatives of Genus One and Three Knots(2017-04-20) Park, Junghwan; Harvey, ShellyA derivative L of an algebraically slice knot K is an oriented link disjointly embedded in a Seifert surface of K such that its homology class forms a basis for a metabolizer H of K. For genus one knots, we produce a new example of a smoothly slice knot with non-slice derivatives. Such examples were first discovered by Cochran and Davis. In order to do so, we define an operation on a homology B^4 that we call an n-twist annulus modification. Further, we give a new construction of smoothly slice knots and exotically slice knots via n-twist annulus modifications. For genus three knots, we show that the set S_{K,H} ={ mu_L(123) - mu_L'(123) | L,L' are derivatives associated with a metabolizer H} contains n · Z, where n is an integer determined by a Seifert form of K and a metabolizer H. As a corollary, we show that it is possible to realize any integer as the Milnor's triple linking number of a derivative of the unknot on a fixed Seifert surface and with a fixed metabolizer.Item Extensions of the Fox-Milnor Condition(2022-04-22) Williams, Shawn; Harvey, ShellyThe search for slice knots is an important task in low dimensional topology. In the 1960s, Fox and Milnor proved a theorem stating that the Alexander polynomial of a slice knot satisfies a special factorization. A decade later, Kawauchi extended this theorem for the multivariable Alexander polynomial of slice links. This factorization, known as the Fox-Milnor condition, has been used and generalized many times as an obstruction to a link being slice. In this defense, we will see two more extensions of this condition, first to the multivariable Alexander polynomial of 1-solvable links, and then for the first order Alexander polynomial of ribbon knots.Item Filtering smooth concordance classes of topologically slice knots(msp, 2013) Cochran, Tim D.; Harvey, Shelly; Horn, PeterWe propose and analyze a structure with which to organize the difference between a knot in S3 bounding a topologically embedded 2–disk in B4 and it bounding a smoothly embedded disk. The n–solvable filtration of the topological knot concordance group, due to Cochran–Orr–Teichner, may be complete in the sense that any knot in the intersection of its terms may well be topologically slice. However, the natural extension of this filtration to what is called the n–solvable filtration of the smooth knot concordance group, is unsatisfactory because any topologically slice knot lies in every term of the filtration. To ameliorate this we investigate a new filtration, fBng, that is simultaneously a refinement of the n–solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. We show that each Bn=BnC1 has infinite rank. But our primary interest is in the induced filtration, fTng, on the subgroup, T , of knots that are topologically slice. We prove that T =T0 is large, detected by gauge-theoretic invariants and the , s , –invariants, while the nontriviality of T0=T1 can be detected by certain d –invariants. All of these concordance obstructions vanish for knots in T1 . Nonetheless, going beyond this, our main result is that T1=T2 has positive rank. Moreover under a “weak homotopy-ribbon” condition, we show that each Tn=TnC1 has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity.Item First Order Signatures and Knot Concordance(2012-09-05) Davis, Christopher; Cochran, Tim D.; Harvey, Shelly; Borcea, LilianaInvariants of knots coming from twisted signatures have played a central role in the study of knot concordance. Unfortunately, except in the simplest of cases, these signature invariants have proven exceedingly difficult to compute. As a consequence, many knots which presumably can be detected by these invariants are not a well understood as they should be. We study a family of signature invariants of knots and show that they provide concordance information. Significantly, we provide a tractable means for computing these signatures. Once armed with these tools we use them first to study the knot concordance group generated by the twist knots which are of order 2 in the algebraic concordance group. With our computational tools we can show that with only finitely many exceptions, they form a linearly independent set in the concordance group. We go on to study a procedure given by Cochran-Harvey-Leidy which produces infinite rank subgroups of the knot concordance group which, in some sense are extremely subtle and difficult to detect. The construction they give has an inherent ambiguity due to the difficulty of computing some signature invariants. This ambiguity prevents their construction from yielding an actual linearly independent set. Using the tools we develop we make progress to removing this ambiguity from their procedure.Item Higher-Order Signature Cocycles for Subgroups of Mapping Class Groups and Homology Cylinders(Oxford University Press, 2012) Cochran, Tim D.; Harvey, Shelly; Horn, Peter D.Item Infection By A String Link(2015-04-23) Vela, Diego; Hassett, Brendan; Harvey, Shelly; Cox, Steve; Leidy, ConstanceSatellite constructions on a knot can be thought of as taking some strands of a knot and then tying in another knot. Using satellite constructions one can construct many distinct isotopy classes of knots. Pushing this further one can construct distinct concordance classes of knots which preserve some algebraic invariants. Infection is a generalization of satellite operations which has been previously studied. An infection by a string link can be thought of as grabbing a knot at multiple locations and then tying in a link. Cochran, Friedl and Teichner showed that any algebraically slice knot is the result of infecting a slice knot by a string link(1). In this paper we use the infection construction to show that there exist knots which arise from infections by n-component string links that cannot be obtained by (n − 1)-component string links.Item Lower order solvability of links(2013-09-16) Martin, Taylor; Harvey, Shelly; Cochran, Tim D.; Goldman, RonThe n-solvable filtration of the link concordance group, defined by Cochran, Orr, and Teichner in 2003, is a tool for studying smooth knot and link concordance that yields important results in low-dimensional topology. We focus on the first two stages of the n-solvable filtration, which are the class of 0-solvable links and the class of 0.5-solvable links. We introduce a new equivalence relation on links called 0-solve equivalence and establish both an algebraic and a geometric characterization 0-solve equivalent links. As a result, we completely characterize 0-solvable links and we give a classification of links up to 0-solve equivalence. We relate 0-solvable links to known results about links bounding gropes and Whitney towers in the 4-ball. We then establish a sufficient condition for a link to be 0.5-solvable and show that 0.5-solvable links must have vanishing Sato-Levine invariants.Item Lower Order Solvability, Seifert Forms, and Blanchfield Forms of Links(2019-04-18) Seger, Sarah; Harvey, ShellyWe define and study specific generalizations of Seifert forms and Blanchfield forms to links and study their relationships with lower order solvability and with each other. We define Seifert Z-surfaces for links with pairwise linking numbers zero and prove that if a link is 0.5-solvable then every Seifert Z-surface has a metabolizer. We use this result to determine that Arf invariants and Milnor's invariants are not sufficient to classify 0.5-solvable links. We define nonsingular localized Blanchfield forms for links with pairwise linking numbers zero and build on work of Cochran-Orr-Teichner and Cochran-Harvey-Leidy to show that 1-solvability implies each of these Blanchfield forms are hyperbolic. We also define Blanchfield forms on the infinite cyclic covers of the exterior of a link with pairwise linking numbers zero and build on work of Friedl-Powell to prove that in a special case, a Seifert Z-surface having a metabolizer implies the Blanchfield form is hyperbolic. There are well known definitions of boundary Seifert surfaces and multivariable Blanchfield forms for boundary links. We define a boundary metabolizer for a boundary Seifert surface, which is more restrictive than the usual definition of a metabolizer, and prove that the existence of a boundary metabolizer implies both 0.5-solvability and that the multivariable Blanchfield form is hyperbolic.Item Obstructions to the Concordance of Satellite Knots(2012-09-05) Franklin, Bridget; Cochran, Tim D.; Harvey, Shelly; Scott, David W.Formulas which derive common concordance invariants for satellite knots tend to lose information regarding the axis a of the satellite operation R(a,J). The Alexander polynomial, the Blanchfield linking form, and Casson-Gordon invariants all fail to distinguish concordance classes of satellites obtained by slightly varying the axis. By applying higher-order invariants and using filtrations of the knot concordance group, satellite concordance may be distinguished by determining which term of the derived series of the fundamental group of the knot complement the axes lie. There is less hope when the axes lie in the same term. We introduce new conditions to distinguish these latter classes by considering the axes in higher-order Alexander modules in three situations. In the first case, we find that R(a,J) and R(b,J) are non-concordant when a and b have distinct orders viewed as elements of the classical Alexander module of R. In the second, we show that R(a,J) and R(b,J) may be distinguished when the classical Blanchfield form of a with itself differs from that of b with itself. Ultimately, this allows us to find infinitely many concordance classes of R(-,J) whenever R has nontrivial Alexander polynomial. Finally, we find sufficient conditions to distinguish these satellites when the axes represent equivalent elements of the classical Alexander module by analyzing higher-order Alexander modules and localizations thereof.Item Satellite constructions and topological concordance(2024-03-25) Manchester, Alex; Harvey, ShellyIn [CFT07], Cochran-Friedl-Teicher unified and generalized many existing constructions of topologically slice links using the language of satellite constructions. Many such links have been shown to not be smoothly slice, and links which are topologically but not smoothly slice are some of the most fundamental examples of exotic behavior in 4-dimensional topology. In this thesis, we will give an improvement on the Milnor’s µ¯-invariant condition that appears [CFT07], which will allow us to give some examples of topologically slice links which are not covered by [CFT07]. We will then move on to prove an approximate relativization of this theorem, and then show that a wide class of metabelian invariants, in particular Casson-Gordon invariants and metabelian ρ-invarinats, do not obstruct the honest relativization from holding. If the honest relativization did hold, it would give strong evidence that knots with homology cobordant 0-surgeries are topologically concordant, which is known to be false smooothly (see [CFHH13] and [Col22]). We will also discuss how topological concordance can be interpreted for links in homology spheres other than S3. While moving to the more general setting of homology spheres does produce new knots and links up to concordance smoothly (see [Don83] and [Lev16]), there is some evidence (see [Dav20a] and [Dav23]) that every knot in a homology sphere is topologically concordant to a knot in S3. We will record the fundamental fact that there is a canonical homology cobordism between any two homology spheres characterized by its simple connectedness, which gives a concrete place to look for concordances.Item Shake Slice and Shake Concordant Links(2017-04-21) Bosman, Anthony Michael; Harvey, ShellyThe study of knots and links up to concordance has proved significant for many problems in low dimensional topology. In the 1970s, Akbulut introduced the notion of shake concordance of knots, a generalization of the study of knot concordance. Recent work of Cochran and Ray has advanced our understanding of how shake concordance relates to concordance, although fundamental questions remain, especially for the class of shake slice knots. We extend the notion of shake concordance to links, generalizing much of what is known for knots, and offer a characterization in terms of link concordance and the infection of a link by a string link. We also discuss a number of invariants and properties of link concordance which extend to shake concordance of links, as well as note several that do not. Finally, we give several obstructions to a link being shake slice.Item Tau invariants of spatial graphs(2016-04-21) Vance, Katherine Rose Poulsen; Harvey, ShellyIn 2003, Ozsvath and Szabo defined the concordance invariant tau for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of tau for knots in S^3 and a combinatorial proof that tau gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in S^3 which extends knot Floer homology. We define a Z-filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O’Donnol’s graph Floer homology. We use this to show that there is a well-defined tau invariant for balanced spatial graphs generalizing the tau knot concordance invariant. In particular, this defines a tau invariant for links in S^3. Using techniques similar to those of Sarkar, we show that our tau invariant gives an obstruction to a link being slice.Item The (n)-Solvable Filtration of the Link Concordance Group and Milnor's mu-Invariaants(2011) Otto, Carolyn Ann; Harvey, ShellyWe establish several new results about the ( n )-solvable filtration, [Special characters omitted.] , of the string link concordance group [Special characters omitted.] . We first establish a relationship between ( n )-solvability of a link and its Milnor's μ-invariants. We study the effects of the Bing doubling operator on ( n )-solvability. Using this results, we show that the "other half" of the filtration, namely [Special characters omitted.] , is nontrivial and contains an infinite cyclic subgroup for links with sufficiently many components. We will also show that links modulo (1)-solvability is a nonabelian group. Lastly, we prove that the Grope filtration, [Special characters omitted.] of [Special characters omitted.] is not the same as the ( n )-solvable filtration.