Meridian of a knot 2026-09-28
A meridian is an oriented simple closed curve on the boundary of a knot exterior that bounds a disk in the removed tubular neighborhood and has linking number one with the knot.
Past exam of the mathematics course of the University of Cambridge 2022 iii Paper 112 5 b Solution 2026-09-28
Distinct components of the positively oriented torus link have linking number one. Since every component also has framing relative to the Seifert framing, the surgery linking matrix and hence the intersection form of the surgery trace areIts Smith normal form is . The surgery exact sequence, equivalently the kernel and cokernel of , gives
In Kirby calculus, sliding the components over one chosen component diagonalizes the framed link to a split -framed unknot and zero-framed unknots. Hencewhere denotes boundary connected sum.
Seifert framing 2026-09-28
The Seifert framing pushes a knot in the direction tangent to a Seifert surface and normal to its boundary. The knot and its push-off then have linking number zero.
Torus link 2026-09-28