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Conway knot and its Piccirillo trace partner
Definition
Let denote the Conway knot represented by the Conway diagram in Figure 1 of Piccirillo's paper, and let denote the knot represented by the partner diagram in Figure 2 of that paper. Choose an orientation on each diagram. The symbols and refer to these specified diagram isotopy classes.
Definition references
Facts & Assumptions
Given: The two diagram isotopy classes specified in the Definition.
Piccirillo labels the knot in Figure 1 as the Conway knot and the knot in Figure 2 as its trace partner . The figures determine the two isotopy classes used here.
Proof
The two isotopy classes are fixed by the two named diagrams; no equivalence of knots is inferred from their polynomial or homological invariants.
The orientations are auxiliary choices for the later oriented invariants; the underlying knot types in this definition are fixed by the diagrams and do not depend on a choice of a different representative in the same ambient isotopy class.
Used by
- Equal zero-traces do not force smooth concordance Counterexample
- Lee computations for the Conway knot and its Piccirillo partner Lemma
- Piccirillo’s Conway diagrams give a dualizable handle link Lemma
- Seifert-matrix certificate for the Conway diagram Lemma
- The Conway knot has normalized Alexander polynomial one Lemma
- Piccirillo’s partner has the same zero-trace and Rasmussen invariant two Theorem
- The Conway knot is topologically slice but not smoothly slice Theorem
Dependency tree · 0 levels
Nothing. This result depends on no other item in the library.
Sources
- Lisa Piccirillo, The Conway knot is not slice (standard reference, not scraped)
- Marc Kegel and Jonathan Spreer, The search for exotic knot traces (standard reference, not scraped)