arXiv subjects

James Conant

Publications and source records attributed to James Conant.

At least 19 recordsLinked to original sources

arXiv2020

Clasper Concordance, Whitney towers and repeating Milnor invariants

We show that for each $k\in\mathbb{N}$, a link $L\subset S^3$ bounds a degree $k$ Whitney tower in the 4-ball if and only if it is \emph{$C_k$-concordant} to the unlink. This means that $L$ is obtained from the unlink by a finite sequence of concordances and degree $k$ clasper surgeries. In our construction the trees associated to the Whitney towers coincide with the trees associated to the claspers. As a corollary to our previous obstruction theory for Whitney towers in the 4-ball, it follows that the $C_k$-concordance filtration of links is classified in terms of Milnor invariants, higher-order Sato-Levine and Arf invariants. Using a new notion of $k$-repeating twisted Whitney towers, we also classify a natural generalization of the notion of link homotopy, called twisted \emph{self $C_k$-concordance}, in terms of $k$-repeating Milnor invariants and $k$-repeating Arf invariants.

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arXiv2016

Addendum to "The Johnson cokernel and the Enomoto-Satoh invariant": The ES-trace detects all top-level partitions

The degree $d$ part of the cokernel $\mathsf C_d$ of the Johnson homomorphism decomposes into irreducible $\mathrm{SP}$-modules indexed by partitions of $d-2r$ for $r\geq 0$: $$\mathsf C_d\cong \mathsf C_d(d)\oplus \mathsf C_d(d-2)\oplus\cdots.$$ In this addendum we calculate $\mathsf{C}_d(d)$ precisely: it is isomorphic to the $\mathrm{GL}(V)$-decomposition of a space of coinvariants $(V^{\otimes d})_{D_{2d}}$, and the isomorphism is induced by Enomoto and Satoh's trace map. This establishes Conjecture 7.2 of the paper "The Johnson Cokernel and the Enomoto-Satoh invariant."

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arXiv2016

The Conway Polynomial and Amphicheiral Knots

According to work of Hartley and Kawauchi in 1979 and 1980, the Conway Polynomial of all negative amphicheiral knots and strongly positive amphicheiral knots factors as $\phi(z)\phi(-z)$ for some $\phi(z)\in\mathbb Z[z]$. Moreover, a 2012 example due to Ermotti, Hongler and Weber shows that this is not true for general amphicheiral knots. On the other hand, in 2006 the first author made a conjecture equivalent to saying that the Conway polynomial of all amphicheiral knots splits as $\phi(z)\phi(-z)$ in the ring $\mathbb Z_4[z]$. In this paper, we establish this conjecture for all periodically amphicheiral knots built from braids, where the period preserves the braid structure. We also give counterexamples to conjectures on the leading coefficient of the Conway polynomial of an amphicheiral knot due to Stoimenow.

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arXiv2015

The Lie Lie algebra

We study the abelianization of Kontsevich's Lie algebra associated with the Lie operad and some related problems. Calculating the abelianization is a long-standing unsolved problem, which is important in at least two different contexts: constructing cohomology classes in $H^k(\mathrm{Out}(F_r);\mathbb Q)$ and related groups as well as studying the higher order Johnson homomorphism of surfaces with boundary. The abelianization carries a grading by "rank," with previous work of Morita and Conant-Kassabov-Vogtmann computing it up to rank $2$. This paper presents a partial computation of the rank $3$ part of the abelianization, finding lots of irreducible $\mathrm{SP}$-representations with multiplicities given by spaces of modular forms. Existing conjectures in the literature on the twisted homology of $\mathrm{SL}_3(\mathbb Z)$ imply that this gives a full account of the rank $3$ part of the abelianization in even degrees.

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arXiv2015

Assembling homology classes in automorphism groups of free groups

The observation that a graph of rank $n$ can be assembled from graphs of smaller rank $k$ with $s$ leaves by pairing the leaves together leads to a process for assembling homology classes for $Out(F_n)$ and $Aut(F_n)$ from classes for groups $\Gamma_{k,s}$, where the $\Gamma_{k,s}$ generalize $Out(F_k)=\Gamma_{k,0}$ and $Aut(F_k)=\Gamma_{k,1}$. The symmetric group $\Sigma_s$ acts on $H_*(\Gamma_{k,s})$ by permuting leaves, and for trivial rational coefficients we compute the $\Sigma_s$-module structure on $H_*(\Gamma_{k,s})$ completely for $k \leq 2$. Assembling these classes then produces all the known nontrivial rational homology classes for $Aut(F_n)$ and $Out(F_n)$ with the possible exception of classes for $n=7$ recently discovered by L. Bartholdi. It also produces an enormous number of candidates for other nontrivial classes, some old and some new, but we limit the number of these which can be nontrivial using the representation theory of symmetric groups. We gain new insight into some of the most promising candidates by finding small subgroups of $Aut(F_n)$ and $Out(F_n)$ which support them and by finding geometric representations for the candidate classes as maps of closed manifolds into the moduli space of graphs. Finally, our results have implications for the homology of the Lie algebra of symplectic derivations.

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arXiv2014

Embedding calculus knot invariants are of finite type

We show that the map on components from the space of classical long knots to the n-th stage of its Goodwillie-Weiss embedding calculus tower is a map of monoids whose target is an abelian group and which is invariant under clasper surgery. We deduce that this map on components is a finite type-(n-1) knot invariant. We also compute the second page in total degree zero for the spectral sequence converging to the components of this tower as Z-modules of primitive chord diagrams, providing evidence for the conjecture that the tower is a universal finite-type invariant over the integers. Key to these results is the development of a group structure on the tower compatible with connect-sum of knots, which in contrast with the corresponding results for the (weaker) homology tower requires novel techniques involving operad actions, evaluation maps, and cosimplicial and subcubical diagrams.

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arXiv2012

Universal quadratic forms and Whitney tower intersection invariants

The first part of this paper exposits a simple geometric description of the Kirby-Siebenmann invariant of a 4--manifold in terms of a quadratic refinement of its intersection form. This is the first in a sequence of higher-order intersection invariants of Whitney towers studied by the authors, particularly for the 4--ball. In the second part of this paper, a general theory of quadratic forms is developed and then specialized from the non-commutative to the commutative to finally, the symmetric settings. The intersection invariant for twisted Whitney towers is shown to be the universal symmetric refinement of the framed intersection invariant. As a corollary we obtain a short exact sequence that has been essential in the understanding of Whitney towers in the 4--ball.

math.GT↗

arXiv2012

Chirality and the Conway polynomial

In recent work with J.Mostovoy and T.Stanford,the author found that for every natural number n, a certain polynomial in the coefficients of the Conway polynomial is a primitive integer-valued degree n Vassiliev invariant, but that modulo 2, it becomes degree n-1. The conjecture then naturally suggests itself that these primitive invariants are congruent to integer-valued degree n-1 invariants. In this note, the consequences of this conjecture are explored. Under an additional assumption, it is shown that this conjecture implies that the Conway polynomial of an amphicheiral knot has the property that C(z)C(iz)C(z^2) is a perfect square inside the ring of power series with integer coefficients, or, equivalently, the image of C(z)C(iz)C(z^2) is a perfect square inside the ring of polynomials with Z_4 coefficients. In fact, it is probably the case that the Conway polynomial of an amphicheiral knot always can be written as f(z)f(-z) for some polynomial f(z) with integer coefficients, and this actually implies the above "perfect squares" conditions. Indeed, by work of Kawauchi and Hartley, this is known for all negative amphicheiral knots and for all strongly positive amphicheiral knots. In general it remains unsolved, and this paper can be seen as some evidence that it is indeed true in general. [Added 2/22/12: Of note is the recent paper arXiv:1106.5634v1 [math.GT] by Ermotti, Hongler and Weber, which finds a counterexample to the conjecture that all Conway polynomials of amphicheiral knots are of the form f(z)f(-z). Intriguingly, their main example still satisfies C(z)C(iz)C(z^2)=f(z)^2 for a power series f(z), making the main conjecture of the present paper that much more compelling, in the author's opinion.]

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arXiv2012

Jacobi identities in low-dimensional topology

The Jacobi identity is the key relation in the definition of a Lie algebra. In the last decade, it also appeared at the heart of the theory of finite type invariants of knots, links and 3-manifolds (and is there called the IHX-relation). In addition, this relation was recently found to arise naturally in a theory of embedding obstructions for 2-spheres in 4-manifolds. We expose the underlying topological unity between the 3- and 4-dimensional IHX-relations, deriving from a picture of the Borromean rings embedded on the boundary of an unknotted genus three handlebody in 3-space. This is most naturally related to knot and 3-manifold invariants via the theory of grope cobordisms.

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arXiv2012

Whitney tower concordance of classical links

This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato-Levine and higher-order Arf invariants, which are obstructions to framing a twisted Whitney tower in the 4-ball bounded by a link in the 3-sphere. Applications include computation of the grope filtration, and new geometric characterizations of Milnor's link invariants.

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arXiv2012

Geometric filtrations of string links and homology cylinders

We show that the Artin representation on concordance classes of string links induces a well-defined epimorphism modulo order n twisted Whitney tower concordance, and that the kernel of this map is generated by band sums of iterated Bing-doubles of any string knot with nonzero Arf invariant. We also continue J. Levine's work [20, 21, 22] comparing two filtrations of the group of homology cobordism classes of 3-dimensional homology cylinders, one defined in terms of an Artin-type representation (the Johnson filtration) and one defined using clasper surgery (the Goussarov-Habiro filtration). In particular, the associated graded groups are completely classified up to an unknown 2-torsion summand for the Goussarov-Habiro filtration, for which we obtain an upper bound, in a precisely analogous fashion to the classification of the Whitney tower filtration of link concordance.

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arXiv2011

Hairy graphs and the unstable homology of Mod(g,s), Out(F_n) and Aut(F_n)

We study a family of Lie algebras {hO} which are defined for cyclic operads O. Using his graph homology theory, Kontsevich identified the homology of two of these Lie algebras (corresponding to the Lie and associative operads) with the cohomology of outer automorphism groups of free groups and mapping class groups of punctured surfaces, respectively. In this paper we introduce a hairy graph homology theory for O. We show that the homology of hO embeds in hairy graph homology via a trace map which generalizes the trace map defined by S. Morita. For the Lie operad we use the trace map to find large new summands of the abelianization of hO which are related to classical modular forms for SL(2,Z). Using cusp forms we construct new cycles for the unstable homology of Out(F_n), and using Eisenstein series we find new cycles for Aut(F_n). For the associative operad we compute the first homology of the hairy graph complex by adapting an argument of Morita, Sakasai and Suzuki, who determined the complete abelianization of hO in the associative case.

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arXiv2011

Milnor Invariants and Twisted Whitney Towers

This paper describes the relationship between the first non-vanishing Milnor invariants of a classical link and the intersection invariant of a twisted Whitney tower. This is a certain 2-complex in the 4-ball, built from immersed disks bounded by the given link in the 3-sphere together with finitely many `layers' of Whitney disks. The intersection invariant is a higher-order generalization of the intersection number between two immersed disks in the 4-ball, well known to give the linking number of the link on the boundary, which measures intersections among the Whitney disks and the disks bounding the given link, together with information that measures the twists (framing obstructions) of the Whitney disks. This interpretation of Milnor invariants as higher-order intersection invariants plays a key role in the classifications of both the framed and twisted Whitney tower filtrations on link concordance (as sketched in this paper). Here we show how to realize the higher-order Arf invariants, which also play a role in the classifications, and derive new geometric characterizations of links with vanishing Milnor invariants of length less than or equal to 2k.

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arXiv2011

Geometric Filtrations of Classical Link Concordance

This paper describes grope and Whitney tower filtrations on the set of concordance classes of classical links in terms of class and order respectively. Using the tree-valued intersection theory of Whitney towers, the associated graded quotients are shown to be finitely generated abelian groups under a (surprisingly) well-defined connected sum operation. Twisted Whitney towers are also introduced, along with a corresponding quadratic enhancement of the intersection theory for framed Whitney towers that measures Whitney-disk framing obstructions. The obstruction theory in the framed setting is strengthened, and the relationships between the twisted and framed filtrations are described in terms of exact sequences which show how higher-order Sato-Levine and higher-order Arf invariants are obstructions to framing a twisted Whitney tower. The results from this paper combine with those in \cite{CST2,CST3,CST4} to give a classifications of the filtrations; see our survey \cite{CST0} as well as the end of the introduction. UPDATE: This paper has been completely subsumed into the paper "Whitney tower concordance of classical links" \cite{WTCCL}.

math.GT↗

arXiv2011

Universal Quadratic Forms and Untwisting Whitney Towers

The first part of this paper completes the classification of Whitney towers in the 4-ball that was started in three related papers. We provide an algebraic framework allowing the computations of the graded groups associated to geometric filtrations of classical link concordance by order n (twisted) Whitney towers in the 4-ball. Higher-order Sato-Levine invariants and higher-order Arf invariants are defined and shown to be the obstructions to framing a twisted Whitney tower. In the second part of this paper, a general theory of quadratic forms is developed and then specialized from the non-commutative to the commutative to finally, the symmetric settings. The intersection invariant for twisted Whitney towers is shown to be the universal symmetric refinement of the framed intersection invariant. UPDATE: The results of the first six sections of this paper have been subsumed into the paper "Whitney tower concordance of classical links."

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arXiv2010

Tree homology and a conjecture of Levine

In his study of the group of homology cylinders, J. Levine made the conjecture that a certain homomorphism eta': T -> D' is an isomorphism. Here T is an abelian group on labeled oriented trees, and D' is the kernel of a bracketing map on a quasi-Lie algebra. Both T and D' have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory, and the homology of the group of automorphisms of the free group. In this paper, we confirm Levine's conjecture. This is a central step in classifying the structure of links up to grope and Whitney tower concordance, as explained in other papers of this series. We also confirm and improve upon Levine's conjectured relation between two filtrations of the group of homology cylinders.

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arXiv2010

Homotopy approximations to the space of knots, Feynman diagrams, and a conjecture of Scannell and Sinha

Scannell and Sinha considered a spectral sequence to calculate the rational homotopy groups of spaces of long knots in n-dimensional Euclidean space, for n greater than or equal to 4. At the end of their paper they conjecture that when n is odd, the terms on the antidiagonal on the second page precisely give the space of primitive Feynman diagrams related to the theory of Vassiliev invariants. In this paper we prove that conjecture. This has the application that the path components of the terms of the Taylor tower for the space of classical long knots are in one-to-one correspondence with quotients of the module of Feynman diagrams, even though the Taylor tower does not actually converge. This provides strong evidence that the stages of the Taylor tower give rise to universal Vassiliev knot invariants in each degree.

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arXiv2010

Boolean formulae, hypergraphs and combinatorial topology

With a view toward studying the homotopy type of spaces of Boolean formulae, we introduce a simplicial complex, called the theta complex, associated to any hypergraph, which is the Alexander dual of the more well-known independence complex. In particular, the set of satisfiable formulae in k-conjunctive normal form with less than or equal to n variables has the homotopy type of Theta(Cube(n,n-k)), where Cube(n,n-k) is a hypergraph associated to the (n-k)-skeleton of an n-cube. We make partial progress in calculating the homotopy type of theta for these cubical hypergraphs, and we also give calculations and examples for other hypergraphs as well. Indeed studying the theta complex of hypergraphs is an interesting problem in its own right.

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