Below are my publications and preprints, in reverse chronological order, along with brief summaries. At times when my website has not been recently updated, a more current list of my papers and preprints may be found by searching my name on the arXiv.
- Cacti, cubes, and framed long knots, with Yongheng Zhang. We show that the operad of projective spineless cacti acts on the Taylor tower for the space of framed 1-dimensional long knots in any Euclidean space. Via an intermediate space of normalized overlapping intervals, we show that this action is compatible with Budney's little 2-cubes action on the space of knots itself at the level of spaces (rather than operads). This implies compatibility of the multiplication and Browder and Dyer-Lashof operations in homology.
- Graphing, homotopy groups of spheres, and spaces of links and knots. I calculated homotopy groups of spaces of long links of various dimensions in a range depending on the dimensions of the sources and the target. The range studied is up to the lowest degree where the third stage of the Taylor tower (or analogues of type-2 Vassiliev invariants) become relevant. A key ingredient was a graphing map that raises source and target dimensions by one. I described generators of these homotopy groups in terms of homotopy groups of spheres and analogues of the trefoil, Whitehead link, and Borromean rings.
- Diagrams for primitive cycles in spaces of pure braids and string links, with Rafal Komendarczyk and Ismar Volic, accepted for publication in Ann. Inst. Fourier. Chen's iterated integrals combined with the Kontsevich formality integral map a certain diagram complex quasi-isomorphically onto the cohomology of spaces of pure braids, as studied in our previous work. Here we studied the dual to this map, showing that it maps homotopy classes injectively into the space of leaf-labeled trivalent trees modulo the Jacobi relations. We also showed that a graphing construction injectively maps the subspace of trees with distinct leaf labels into many spaces of k-dimensional string links.
- Spaces of knots in the solid torus, knots in the thickened torus, and irreducible links in the 3-sphere, with Andrew Havens, Geometriae Dedicata, 2021. We recursively determined the homotopy type of the space of any knot in the solid torus, thus answering a question of Arnold. We did this by determining the homotopy type of irreducible framed links in the 3-sphere. We also obtained homotopy types of spaces of unframed links and spaces of knots in the thickened torus. Our work generalizes previous work of Hatcher and Budney. We aimed to make our account accessible to non-experts, with many examples, pictures, and explicit presentations of fundamental groups.
- Diagram complexes, formality, and configuration space integrals for spaces of braids, with Rafal Komendarczyk and Ismar Volic, Q. J. Math, 2020. We specifed a combinatorial diagram complex which, via Chen's integrals for loop spaces and Kontsevich's formality integrals for configuration spaces, produces all the cohomology of spaces of pure braids in Euclidean spaces of dimension at least 4. We show that these integrals are compatible with the Bott-Taubes integrals for long links, through a map of diagram complexes. In the classical case of ambient dimension 3, the two types of integrals are also compatible and produce Vassiliev invariants of pure braids.
- Bott-Taubes-Vassiliev cohomology classes by cut-and-paste topology, Internat. J. Math., 2019. I recovered a certain integer lattice of all the (Bott-Taubes-)Vassiliev cohomology classes of Cattaneo, Cotta-Ramusino, and Longoni as integer-valued classes. This can be viewed as a rationality result for those classes. The construction also yields mod-p classes which need not be mod-p reductions of integer-valued classes and which could be nontrivial. Though the methods here differ from those in my PhD thesis, I showed here that the methods there yield the same classes.
- Milnor invariants of string links, trivalent trees, and configuration space integrals, with Ismar Volic, Topology Appl., 2019. We showed that the correspondence between trivalent trees and Milnor's link homotopy invariants of string links can be realized by configuration space integrals. This recovered some results of Habegger and Masbaum, but using configuration space integrals instead of the Kontsevich integral. We also produced nontrivial cohomology classes in spaces of links in Euclidean spaces of dimension at least 4.
- Homotopy string links and the kappa-invariant, with Fred Cohen, Rafal Komendarczyk, and Clay Shonkwiler, Bull. Lond. Math. Soc., 2017. We showed that homotopy string links are distinguished by homotopy classes of certain maps of configuration spaces. This proved a string-link analogue of a conjecture of Koschorke. We used a multiplication of maps of configuration spaces which is very similar to the multiplication used in the paper below with Budney, Conant, and Sinha.
- Embedding calculus knot invariants are of finite type, with Ryan Budney, Jim Conant, and Dev Sinha, Algebr. & Geom. Top, 2017. We showed that the Taylor tower for the space of knots (which comes from Goodwillie-Weiss functor calculus) yields finite-type knot invariants. We constructed a homotopy-commutative multiplication on the tower compatible with stacking long knots to show that it yields invariants valued in an abelian group. Habiro's clasper surgery was used to show that they are of finite type. We obtained spectral sequence evidence for a conjecture that the tower is a universal abelian-group-valued finite-type knot invariant. Since then, the conjecture has been proven over the rational numbers by Kosanovic and in a range with mod-p coefficients by Boavida and Horel.
- Homotopy Bott-Taubes integrals and the Taylor tower for the spaces of knots and links, J. Homotopy Related Structures, 2016. This paper originated from the second half of my PhD thesis, with a modification to incorporate the refinement and main result in my paper below on the Milnor triple linking number for string links. The main result was that my construction of homotopy-theoretic configuration space integrals factors through the Taylor tower.
- A colored operad for string link infection, joint with John Burke, Algebr. & Geom. Top., 2015. We generalized Budney's splicing operad from splicing by long knots to infection by long links. We then obtained a decomposition of a subspace of 2-component long links, using the prime decomposition of isotopy classes from our work with Blair below. Subsequently, Batelier and Ducoulombier extended this decomposition to the whole space of 2-component long links by using the Swiss cheese operad.
- A prime decomposition theorem for the 2-string link monoid, with Ryan Blair and John Burke, J. Knot Theory Ramif., 2015. We studied isotopy classes of 2-component long links under the operation of stacking. Unlike for knots, this monoid is nonabelian, but we nonetheless obtained a prime decomposition theorem for it.
- The Milnor triple-linking number of string links by cut-and-paste topology, Algebr. & Geom. Top., 2014. I obtained the Milnor triple linking number for long links via a refinement of the homotopy-theoretic configuration space integrals of my PhD thesis. I later used this refinement to obtain many cohomology classes in spaces of knots and links in Euclidean spaces of dimension at least 4 (see above).
- Configuration space integrals and the cohomology of the space of homotopy string links, with Brian Munson and Ismar Volic, J. Knot Theory Ramif., 2013. We used configuration space integrals and graph complexes to produce cohomology classes in spaces of long embeddings and spaces of long link maps (a.k.a. homotopy string links) of 1-manifolds in Euclidean spaces of dimension at least 4.
- A homotopy-theoretic view of Bott-Taubes integrals and knot spaces, Algebr. & Geom. Top., 2009. This is roughly the first half of my PhD thesis, where I constructed cohomology classes in the space of knots with arbitrary coefficients. The construction even works for arbitrary cohomology theories with respect to which a certain configuration space bundle over the space of knots is orientable. I later showed that the methods here recover the Vassiliev classes of Cattaneo, Cotta-Ramusino, and Longoni (see above).
- Systematic identification of statistically significant network measures, with Etay Ziv, Manuel Middendorf, and Chris Wiggins, Phys. Rev. E, 2005. We developed a new computational method for producing large numbers of features of a network, by considering words up to a certain length in the adjacency matrix and several operations on it.
- Discriminative Topological Features Reveal Biological Network Mechanisms, with Manuel Middendorf, Etay Ziv, Carter Adams, Jen Hom, Chaya Levovitz, Greg Woods, Linda Chen, and Chris Wiggins, BMC Bioinformatics, 2004. We mapped networks into a high-dimensional space using the features from the paper above and then applied machine learning techniques to study networks from real world data, as well as models that simulate them from the literature.
My research is currently supported by NSF grant DMS-2405370.
