# Research

My research interests lie in low-dimensional topology, including 3- and 4-dimensional manifolds, knot and link theory, and the non-commutative algebra used in these fields. In my work, I have used $$L^2$$-invariants (including von Neumann $$\rho$$-invariants) and gauge theoretic tools (including Heegaard Floer homology). Most of my efforts have been concentrated in knot theory — particularly in knot concordance — though some of the tools I have picked up have yielded nice results in mapping class groups.

## Publications and preprints

11. On the intersection ring of graph manifolds
with Margaret Doig
to appear in Transactions of the American Mathematical Society
Available at arXiv: 1412.3990

10. On computing the first higher-order Alexander modules of knots
Experimental Mathematics, 23 (2014), no. 2, 153–169, doi:10.1080/10586458.2014.882806.
Available through the publisher or at arXiv: 1303.1545

9. Structure in the bipolar filtration of topologically slice knots
joint with Tim Cochran
Algebraic & Geometric Topology 15 (2015), 415-428, doi:10.2140/agt.2015.15.415.
available through the publisher or at arXiv: 1208.5788

8. Filtering smooth concordance classes of topologically slice knots
joint with Tim Cochran and Shelly Harvey
Geometry & Topology, 17 (2013), no. 4, 2103–2162, doi:10.2140/gt.2013.17.2103.
Available through the publisher or at arXiv: 1201.6283

7. Knot concordance and homology cobordism
joint with Tim Cochran, Bridget Franklin and Matt Hedden
Proceedings of the American Mathematical Society, 141 (2013), no. 6, 2193–2208.
Available through the publisher or at arXiv: 1102.5730

6. Higher-order Signature Cocycles for Subgroups of Mapping Class Groups and Homology Cylinders
joint with Tim Cochran and Shelly Harvey
International Mathematics Research Notices, 2012, no. 14, 3311-3373, doi:10.1093/imrn/rnr149.
Available through the publisher or at arXiv: 1003.4977

5. A Higher-order Genus Invariant and Knot Floer Homology
Proceedings of the American Mathematical Society, 138, (2010), 2209-2215.
Available through the publisher or at arXiv: 0901.2095

4. Higher-order Analogues of the Slice Genus of a Knot
International Mathematics Research Notices, 2011, no. 5, 1091-1106.
Available through the publisher or at arXiv: 0807.0434

3. The Non-triviality of the Grope Filtrations of the Knot and Link Concordance Groups
Commentarii Mathematici Helvetici, 85, (2010), no. 4, 751-773.
Available through the publisher or at arXiv: 0804.2661