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Kalman Lab
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A lab that studies knots and three-dimensional shapes using topology and combinatorics.
AboutAbout the lab
Knots and three-dimensional space through topology
The Kalman Lab studies low-dimensional geometric topology, especially contact 3-manifolds and Legendrian knots.
Classical knot invariants such as the HOMFLY polynomial are also among its subjects of study.
The lab also works on the combinatorics of graphs and lattice polytopes, and has found a new connection between quantum knot polynomials and Floer homology.
Recently, it has also become interested in mathematical models of soft matter physics.
Key terms
- Contact 3-manifold
- A three-dimensional space equipped with a structure (a contact structure) in which a plane is assigned to each point in a twisting manner
- Legendrian knot
- A knot placed in space so as to follow the rules of the contact structure
- Lattice polytope
- A polytope whose vertices all lie at points with integer coordinates (lattice points)
ResearchResearch themes
The lab's main subject is low-dimensional geometric topology, including contact 3-manifolds, Legendrian knots, and classical knot invariants. It also works on the combinatorics of graphs, hypergraphs, and lattice polytopes.
*Illustrative imageLegendrian knots and contact structures
The lab studies contact 3-manifolds, Legendrian knots, and their Floer homology. It constructed the first example of a loop of Legendrian knots that is trivial as a loop of smooth knots yet essential as a loop of Legendrian knots.
*Illustrative imageCombinatorics of graphs and lattice polytopes
The lab introduced a new polynomial invariant for integer polymatroids that generalizes the Tutte polynomial. It also studies h*-vectors of lattice polytopes and subdivisions of root polytopes of graphs.
*Illustrative imageMathematical models of soft matter physics
Recently, the lab has also become interested in continuous and discrete mathematical models that arise in soft matter physics.
MembersFaculty
- Associate ProfessorTamas KalmanPublications ↗
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Trefoil knot build with trebl torus type construction set bricks” Dutchdudedesign / CC BY-SA 4.0 (cropped)
- Legendrian knots and contact structures: “Standard contact structure r 3” Pmassot / CC BY-SA 3.0 (cropped)
- Combinatorics of graphs and lattice polytopes: “Permutohedron in simplex of order 3 (plain)” Watchduck / CC BY-SA 4.0 (cropped)
- Mathematical models of soft matter physics: “Nematic liquid crystal” Ingo Dierking / CC BY-SA 4.0 (cropped)
- Related lab (五味研究室): “Klein bottle glass2” PaulT (Gunther Tschuch) / CC BY-SA 4.0 (cropped)
- Related lab (谷田川研究室): “Riemann surface sqrt” Jan Homann / Public domain (cropped)
- Related lab (内藤研究室): “E8 edgeoverlap” Tom Ruen / Public domain (cropped)
- Related lab (大矢研究室): “E8Petrie” Jgmoxness / CC BY-SA 3.0 (cropped)
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