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- Mentors: Ryan Stees and Vinicius Ambrosi
- Faculty coordinator: Prof. Dylan Thurston
A knot in the 3-sphere is said to be fibered if there exists a fibration of the knot complement over the circle. Exteriors of fibered knots may be expressed as fiber bundles over the circle, with fiber a Seifert surface for the knot. The Milnor fibration may be used to show all torus knots are fibered. This fibration gives a particularly nice way of putting coordinates on the torus knot exterior, and we can view individual fibers in Euclidean 3-space via stereographic projection. In his book “Knots and Links”, Rolfsen gives an alternative description of the trefoil knot exterior as a fiber bundle over the circle (See, Figure 3a). This description is generalizable and, presumably, related to the Milnor fibration.
A knot in the 3-sphere is said to be fibered if there exists a fibration of the knot complement over the circle. Exteriors of fibered knots may be expressed as fiber bundles over the circle, with fiber a Seifert surface for the knot. The Milnor fibration may be used to show all torus knots are fibered. This fibration gives a particularly nice way of putting coordinates on the torus knot exterior, and we can view individual fibers in Euclidean 3-space via stereographic projection. In his book “Knots and Links”, Rolfsen gives an alternative description of the trefoil knot exterior as a fiber bundle over the circle (See, Figure 3a). This description is generalizable and, presumably, related to the Milnor fibration.
The goal of this project is to understand the Milnor and Rolfsen fibrations and to 3D-print some of their fibers, for example Figure 3b.
The goal of this project is to understand the Milnor and Rolfsen fibrations and to 3D-print some of their fibers, for example Figure 3b.
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