Browsing by Author "Cochran, Tim D."
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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 Approximation of knot invariants by Vassiliev invariants(1995) Sirotine, Serguei A.; Cochran, Tim D.We give a criterion for a knot invariant, which is additive under connected sum, to be approximated by a sequence of Vassiliev (finite-type) invariants. This partially answers the question: can an arbitrary knot invariant be approximated by Vassiliev invariants? A knot invariant, which is additive under connected sum, is approximated by a sequence of Vassiliev invariants if and only if for any knot K it is constant on the infinite intersection ${\cap}K(n)$. Here K(n) is the set of knots whose class in the group GK$\rm\sb{n}$ of the classes of n-equivalent knots (due to Gusarov) differs from the class of K by some torsion element of GK$\rm\sb{n}$. Roughly speaking, ${\cap}K(n)$ is the set of knots which cannot be distinguished from K by any Vassiliev invariant. Thus, it is impossible to solve the problem of approximating a knot invariant by Vassiliev invariants without answering the question which knots can be separated by Vassiliev invariants. It is also shown that generalized signatures and certain Minkowski units are not Vassiliev invariants.Item Bordism invariants of the mapping class group(2004) Heap, Aaron; Cochran, Tim D.We define new bordism and spin bordism invariants of certain subgroups of the mapping class group of a surface. In particular, they are invariants of the Johnson filtration of the mapping class group. The second and third terms of this filtration are the well-known Torelli group and Johnson subgroup, respectively. We introduce a new representation in terms of spin bordism, and we prove that this single representation contains all of the information given by the Johnson homomorphisms, the Birman-Craggs homomorphisms, and the Morita homomorphisms.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 Divisibility of the Conway polynomial of links(2001) Lampazzi, Amy M. Noel; Cochran, Tim D.The Conway polynomial ∇K = c0 + c1z + c2z2...of a link K is an invariant of links. In this paper we extend a theorem of J. Levine [5] regarding divisibility of the Conway polynomial by monomials of the form zi. Three different definitions of finite type invariants of links are presented, including a definition of surgery finite type. When the coefficients ci are considered as finite type invariants, the type of these invariants is closely related to the degree of the monomials which can be factored out of ∇K. With this in mind, we prove an extension of a conjecture by T. Cochran and P. Melvin [2] concerning the divisibility of an alternating sum sum S<L ∇K(Sigma S) of Conway polynomials of an algebraically split link K in various surgered spheres. This result was also proved for a more general case in which the link K is not necessarily algebraically split. Finally, corollaries relate these theorems to the type of the coefficients ci, considered as finite type invariants.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 analogues of genus and slice genus of classical knots(2009) Horn, Peter Douglas; Cochran, Tim D.We define invariants analogous to the genus and slice genus of knots in S3. For algebraically slice, genus one knots, we define the differential genus, denoted dg, and we prove it is independent of the Alexander polynomial and knot Floer homology. For knots which bound Gropes of height n + 2 in D 4, we define the nth-order genus, denoted gn. Each of the n th-order genera is a generalization of the slice genus. For each n ≥ 1, we construct knots with identical lower-order genera and distinct nth-order genera, thus proving that these invariants are independent of one another. Finally, we employ the higher-order genera to give a refinement of the Grope filtration of the knot concordance group.Item Higher-order linking forms(2004) Leidy, Constance; Cochran, Tim D.Trotter [T] found examples of knots that have isomorphic classical Alexander modules, but non-isomorphic classical Blanchfield linking forms. T. Cochran [C] defined higher-order Alexander modules, An , (K), of a knot, K, and higher-order linking forms, Bℓn (K), which are linking forms defined on An , (K). When n = 0, these invariants are just the classical Alexander module and Blanchfield linking form. The question was posed in [C] whether Trotter's result generalized to the higher-order invariants. We show that it does. That is, we construct examples of knots that have isomorphic nth-order Alexander modules, but non-isomorphic nth-order linking forms. Furthermore, we define new higher-order linking forms on the Alexander modules for 3-manifolds considered by S. Harvey [H]. We construct examples of 3-manifolds with isomorphic nth-order Alexander modules, but non-isomorphic nth-order linking forms.Item Higher-order polynomial invariants of 3-manifolds giving lower bounds for Thurston norm(2002) Harvey, Shelly Lynn; Cochran, Tim D.We define a new infinite sequence of invariants, d&d1;n for n ≥ 0, of a group G that measure the size of the successive quotients of the derived series of G. In the case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold invariants. These invariants are closely related to the topology of the 3-manifold. We show that they give lower bounds for the Thurston norm. Moreover, we show that they give better estimates for the Thurston norm than the previously known bounds given by the Alexander norm, d&d1;0 . To do this, we exhibit 3-manifolds whose Alexander norm is trivial but whose d&d1;n are strictly increasing and can be made arbitrarily large. Other applications are made to detecting 3-manifolds that fiber over S 1 and to detecting 4-manifolds that admit no symplectic structure.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 Homology boundary links, patterns, and Seifert forms(1996) Bellis, Paul Andrew; Cochran, Tim D.Homology boundary links have become an increasingly important class of links, largely due to their significance in the ongoing concordance classification of links. Tim Cochran and Jerome Levine defined an algebraic object called a pattern, associated to an homology boundary link, which can be used to study the deviance of an homology boundary link from being a boundary link. Since a pattern is a set of m elements which normally generates the free group of rank m, any invariants which detect non-trivial patterns can be applied to the purely algebraic question of when such a set is a set of conjugates of a generating set for the free group. This thesis contains two major results. First, we will give a constructive geometric proof that all patterns are realized by some ribbon homology boundary link $\rm L\sp{n}$ in $\rm S\sp{n+2}$ We shall also prove an analogous existence theorem for calibrations of ${\rm I\!E}$-links, a more general and less understood class of links than homology boundary links. Second, we will prove that given a boundary link L and Seifert system V for L admitting pattern $\rm P\sb{L}$, the strong fusion of L along multiple fusion bands, denoted SF(L), is an homology boundary link possessing particular generalized Seifert system Y admitting specific pattern $\rm P\sb{SF(L)}$.Item Homology cobordism and Seifert fibered 3-manifolds(American Mathematical Society, 2014) Cochran, Tim D.; Tanner, DanielIt is known that every closed oriented 3-manifold is homology cobordant to a hyperbolic 3-manifold. By contrast we show that many homology cobordism classes contain no Seifert fibered 3-manifold. This is accomplished by determining the isomorphism type of the rational cohomology ring of all Seifert fibered 3-manifolds with no 2-torsion in their first homology. Then we exhibit families of examples of 3-manifolds (obtained by surgery on links), with fixed linking form and cohomology ring, that are not homology cobordant to any Seifert fibered space (as shown by their rational cohomology rings). These examples are shown to represent distinct homology cobordism classes using higher Massey products and Milnor's µ-invariants for links.Item Homomorphic images of link quandles(2004) Wallace, Steven D.; Cochran, Tim D.We study the difference between quandles that arise from conjugation in groups and those which do not. As a result, we define conjugation subquandles, and show that not all quandles or keis are in this class of examples. We investigate coloring by keis which are not conjugation subquandles. And we investigate the relationship between decomposable quandles or keis and link colorings. Subsequently, we analyze what kinds of quandles or keis can be homomorphic images of a knot quandle.Item Invariants of graphs(1996) Ghuman, Simrat M.; Cochran, Tim D.We address a classical problem in low dimensional topology: the classification of tamely embedded, finite, connected graphs $G$ in $S\sp3$ up to ambient isotopy. In the case that the graph $G$ is homeomorphic to $S\sp1$, our problem reduces to the embedding problem for knots in $S\sp3$. Our major result is the existence of a unique isotopy class of longitudes of a cycle for an infinite class of graphs. We then define new invariants for this infinite class of graphs. First we define a longitude $l\sb{c}$ of a cycle $c$ in $G$. In contrast to the situation of a knot, for a graph it is quite difficult to canonically select an isotopy class of longitudes, since the mapping class group of a many punctured torus is very large. However we prove that longitudes exist for any cycle in any finite graph and are unique in $H\sb1(S\sp3-G;\doubz)$. This definition of a longitude can be considered an extension of the definition of a longitude of a tamely embedded knot in $S\sp3$. We describe the specific conditions under which $l\sb{c}$ is unique in $\Pi$, the fundamental group of the graph complement, as well as the class of graphs which possess a basis of unique longitudes. Next, in the situation in which $l\sb{c}$ is unique for a cycle $c$ in $G$, we define a sequence of invariants $\bar\mu\sb{G}$ which detects whether $l\sb{c}$ lies in $\Pi\sb{n}$, the $n\sp{th}$ term of the lower central series of $\Pi$. These invariants can be viewed as extensions of Milnor's $\bar\mu\sb{L}$ invariants of a link $L$. Although $\bar\mu\sb{G}$ is not a complete invariant, we provide an example illustrating that $\bar\mu\sb{G}$ is more sensitive than Milnor's $\bar\mu\sb{L}$, where L is the subgraph of G consisting of a link.Item Invariants of graphs(1993) Ghuman, Simrat M.; Cochran, Tim D.We define an infinite sequence of invariants $\bar\mu\sb{K}$ of connected, finite graphs K. These invariants detect whether or not a "longitude" associated to a cycle in K lies in the $n\sp{th}$ term of the lower central series of $\pi\sb1(S\sp3-K,p)$. In certain cases, these invariants can be compared to Milnor's $\bar\mu$-invariants associated to links contained in K, and are found to be more discriminating.Item Knot Concordance and Homology Cobordism(American Mathematical Society, 2013-06) Cochran, Tim D.; Franklin, Bridget D.; Hedden, Matthew; Horn, Peter D.We consider the question: “If the zero-framed surgeries on two oriented knots in S3 are Z-homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?” We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on K is Z-homology cobordant to the zero-framed surgery on many of its winding number one satellites P(K). Then we prove that in many cases the τ and s-invariants of K and P(K) differ. Consequently neither τ nor s is an invariant of the smooth homology cobordism class of the zero-framed surgery. We also show that a natural rational version of this question has a negative answer in both the topological and smooth categories by proving similar results for K and its (p, 1)-cables.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 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 State cycles, quasipositive modification, and constructing H-thick knots in Khovanov homology(2010) Elliott, Andrew; Cochran, Tim D.We study Khovanov homology classes which have state cycle representatives, and examine how they interact with Jacobsson homomorphisms and Lee's map phi. As an application, we describe a general procedure, quasipositive modification, for constructing H-thick knots in rational Khovanov homology. Moreover, we show that specific families of such knots cannot be detected by Khovanov's thickness criteria. We also exhibit a sequence of prime links related by quasipositive modification whose width is increasing.