Research

I am interested in low-dimensional topology and geometry, with a focus on knot theory. My research up to now has been centered around knot homologies, quantum invariants, and their applications. I am currently working (also in collaboration) on various projects concerning Khovanov-Rozansky sl(N)-homologies and their deformations, the study of knot and link concordance from different perspectives, and virtual knots. Here is a small list of some topics I am interested in.

  1. Properties of Khovanov-Rozansky homologies and their deformations;
  2. Concordance of knots and links, obstructions to sliceness, slice genus computations and slice-torus invariants;
  3. Computability and categorification of quantum invariants for links and 3-manifolds;
  4. Invariants for surfaces in four dimensional spaces;
  5. Invariants for graphs in three dimensional spaces;
  6. Invariants for transverse knots and links arising from link homologies, their properties, and their applications to the study of braids;
  7. Link homologies for virtual knots, and applications to the study of virtual concordance;
  8. Spacification of link homologies and their properties.

Papers:

(With P. Lisca) Symmetric union diagrams and refined spin models, Canadian Mathematical Bulletin, DOI 10.4153/S0008439518000115 (arXiv:1804.09157)

A Bennequin-type inequality and combinatorial bounds (arXiv:1707.03424, To appear in the Michigan Mathematical Journal)

Transverse invariants from Khovanov-type homologies, J. Knot Theory Ramifications 28 (2019), no. 1, 1950012, 37 pp., DOI 10.1142/S0218216519500123 (arXiv:1705.03481).

(With A. Cavallo) Slice-torus concordance invariants and Whitehead doubles of links (arXiv:1806.10358, To appear in the Canadian Journal of Mathematics)

Pre-prints:

(With P. Lisca) On symmetric equivalence of symmetric union diagrams (arXiv:1901.10270, Submitted)

Transverse invariants from the deformations of Khovanov sl3-homology (arXiv:1806.00752, Submitted)

Ph.D. Thesis:

Transverse invariants from the deformations of Khovanov sl2- and sl3-homologies

E-Mail Addresses:
carlo [dot] collari [dot] math[at]gmail [dot] com
cc6706[at] nyu [dot] edu