Alphabeta MathHow statement and proof provenance work
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Smooth zero-trace of a knot
Definition
Let be a smooth oriented knot with a chosen tubular neighborhood and preferred longitude . Its smooth zero-trace is the compact smooth -manifold obtained from by attaching a single -handle to along , using the framing whose parallel curve represents ; corners are rounded.
Definition references
Facts & Assumptions
Given: A smooth oriented knot , a chosen tubular neighborhood, and its preferred longitude.
Meridian and preferred longitude: the preferred longitude is the degree-one peripheral curve whose class is zero in the first homology of the knot exterior.
Proof
The attaching framing determines a parallel curve on the boundary torus of the chosen tubular neighborhood. By [F1], the zero framing is the one whose parallel represents the preferred longitude, which is the stated attaching convention.
The construction uses the supplied preferred-longitude representative as attaching data; it makes no uniqueness claim about such representatives and does not change the chosen attachment.
Depends on
Used by
- The unknot zero-trace and a dualizable cancellation diagram Example
- Piccirillo’s Conway diagrams give a dualizable handle link Lemma
- A dualizable handle link gives two zero-trace descriptions Theorem
- A knot is smoothly slice exactly when its zero-trace embeds in S⁴ Theorem
- Piccirillo’s partner has the same zero-trace and Rasmussen invariant two Theorem
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Lisa Piccirillo, The Conway knot is not slice (standard reference, not scraped)
- Jennifer Hom, Getting a handle on the Conway knot (standard reference, not scraped)