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Murakami Lab
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A topology lab that studies the shapes of knots and 3-manifolds.
AboutAbout the lab
Reading the shapes of knots and spaces through mathematics.
The Murakami Lab studies knot theory and 3-manifolds, which are fields within topology.
The lab investigates the shapes of knots and 3-dimensional spaces using algebraic tools.
In particular, it uses "quantum invariants" built from quantum groups to study the shapes of knots and their relationship to hyperbolic volume.
Key terms
- Knot theory
- A branch of mathematics that classifies knots in closed loops of string, and studies their properties, without untying them
- Quantum invariant
- A number determined by the shape of each knot or 3-manifold, computed using a tool called quantum groups
- Hyperbolic volume
- The volume determined when hyperbolic geometry is applied to the space surrounding a knot
ResearchResearch themes
The Murakami Lab studies topology, especially knot theory and 3-manifolds. Currently, it investigates the relationship between quantum invariants of knots and hyperbolic volume.
*Illustrative imageVolume Formula for Tetrahedra
The lab derived a formula for the volume of a tetrahedron in the 3-sphere.
*Illustrative image6j-symbols and Hyperbolic Volume
The lab computed 6j-symbols built from representations of quantum groups and clarified their relationship to the volume of hyperbolic tetrahedra.
*Illustrative imageDefining Knot Invariants
Using the structure of a special quantum group, the lab defined a new invariant for distinguishing knots and investigated its relationship to other invariants.
*Illustrative imageColored Alexander Invariant
The lab redefined a previously defined knot invariant using a tool called the universal R-matrix.
MembersFaculty
- ProfessorJun Murakami
VoicesStudent voices
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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)
- Volume Formula for Tetrahedra: “Tetrahedrite - Black Pine mine, Granite Co., Montana, USA” Ivar Leidus / CC BY-SA 4.0 (cropped)
- 6j-symbols and Hyperbolic Volume: “H center ultrawide” Roice3 / CC BY-SA 3.0 (cropped)
- Defining Knot Invariants: “TekIpteCifteSarmalliDugme” Cobanyastigi / CC0 (cropped)
- Colored Alexander Invariant: “Shimenawa - Meiji Shrine” Stéfan Le Dû from Nantes, France / CC BY-SA 2.0 (cropped)
- Related lab (Ikeda Lab): “Mathematisches-Gipsmodell.Fläche3.Ordnung - Kopie” Museopedia / CC BY-SA 4.0 (cropped)
- Related lab (Usuba Lab): “CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique” Jérémy Barande / CC BY-SA 3.0 (cropped)
- Related lab (Ogawa Lab): “Chladni plate 14” Exhibit made by Estes Objethos Atelier, photo by Rodrigo Tetsuo Argenton / CC BY-SA 4.0 (cropped)
- Related lab (Ozaki Lab): “Minoritenkirche rose window - Vienna” MrPanyGoff / CC BY-SA 3.0 (cropped)
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