# Christopher Davis, Ph.D.

My research interests are in low dimensional topology:

•  3-dimensional topology: knot and links in S3 (or other 3-manifolds), S-equivalence, homotopy.
•  4-dimensional topology: knot and link concordance, homology cobordism.

My joint projects with undergraduate students include the topics of link theory, C-complexes, and Milnor's triple linking number

###### Research and Creative Activities

Low dimensional topology including:

• S-equivalence, homotopy, C-complexes
• Concordance
• 3-manifolds
• knot theory in 3-manifolds
• homology cobordism
###### Education
• Ph.D., Rice University (Mathematics)
• B.A. Westminster College of Salt Lake City
###### Published Research

• The C-complex clasp number of links Joint with Jonah Amundsen, Eric Anderson, and Daniel Guyer. Rocky Mountain J. Math., Volume 50, Number 3 (December 2020), 839-850. DOI: 10.1216/rmj.2020.50.839
• On the indeterminacy of Milnor’s triple linking number. Joint with Jonah Amundsen and Eric Anderson. Journal of Knot Theory and its Ramifications. Vol. 29, No. 09, 2050064 (August 2020). https://doi.org/10.1142/S0218216520500649

•  When do links have equivalent C-complexes? Joint with Grant Roth. Journal of Knot Theory and its Ramifications. (January 2017) DOI: 10.1142/S0218216517500109

Other publications include:

• The relative Whitney trick and its applications. Joint with Patrick Orson and JungHwan Park. Selecta Mathematica. (December 2021) DOI: https://doi.org/10.1007/s00029-021-00738-y
• Moves relating C-complexes: A correction to Cimasoni’s “A geometric construction of the Conway potential function.” Joint with Taylor Martin and Carolyn Otto. Journal of Topology and its Applications. (October 2021) DOI: https://doi.org/10.1016/j.topol.2021.107799
• Linear independence of cables in the knot concordance group. Joint with JungHwan Park and Arunima Ray. Transactions of the American Mathematical Society (February 2021) DOI: https://doi.org/10.1090/tran/8336
• Triple linking numbers and surface systems. Joint with Matthias Nagel, Patrick Orson, and Mark Powell. Indiana University Journal of Mathematics. (December 2020). DOI: 10.1512/iumj.2020.69.8081
• Concordance, crossing changes, and knots in homology spheres. (December 2019) Canadian Mathematical Bulletin. DOI: https://doi.org/10.4153/S0008439519000791
• Concordance to links with an unknotted component.  Joint with JungHwan Park.   Mathematical Proceedings of the Cambridge Philosophical Society.  (October 2019) DOI: https://doi.org/10.1017/S0305004119000367.
• Topological concordance of knots in homology spheres and the solvable filtration. (December 2019) Journal of Topology.  DOI: 10.1112/topo.12126
• Every genus 1 algebraically slice knot is $1$-solvable.  Joint with Taylor Martin, Carolyn Otto, and JungHwan Park. Transactions of the American Mathematical Society. (May 2019) https://doi.org/10.1090/tran/7682
• Concordance of knots in $S^1\times S^2$. Joint with Matthias Nagel, JungHwan Park, and Arunima Ray. Journal of the London Mathematical Society. (March 2018) DOI: 10.1112/jlms.12125

•  A new family of links topologically, but not smoothly, concordant to the Hopf link.  Joint with Arunima Ray.  Journal of Knot Theory and its Ramifications.  (February 2017)  DOI: 10.1142/S0218216517400028

•  Satellite operations as a group action on knot concordance.  Joint with Arunima Ray.  Algebraic and Geometric Topology 16-2 (April 2016), 945--969.  DOI 10.2140/agt.2016.16.945

• Counterexamples to Kauffman's conjecture on Slice knots. Joint with Tim Cochran. Advances in Mathematics ( April 2015)  DOI: 10.1016/j.aim.2014.12.006

• Injectivity of satellite operators in knot concordance.  Joint with Tim Cochran and Arunima Ray Journal of Topology (April 2014)  DOI: 10.1112/jtopol/jtu003

•  Linear independence of knots arising from iterated infection without the use of the Tristram-Levine signature. International Mathematics Research Notices.  (January 2013)  DOI: 10.1093/imrn/rns277

• Von Neumann rho invariants and torsion in the topological knot concordance group.  Algebraic and Geometric Topology.  (April 2012) DOI: 10.2140/agt.2012.12.753

• Strong coprimality and strong irreducibility of Alexander polynomials. joint with Evan M.~Bullock.  Topology and its Applications.  (January 2012)  DOI: 10.1016/j.topol.2011.08.019