Smooth and topological slice genus

Depends on

• Nonemptiness and minima for the smooth and locally flat slice genera
• Seifert surfaces and smooth knot genus
• Every smooth oriented knot in $S^3$ bounds a Seifert surface
• Locally flat proper surfaces and concordance collars
• Oriented links in the three-sphere and ambient isotopy
• The Axiom of Countable Choice ($\mathrm{AC}_\omega$)

Used by

• Vanishing tau does not imply smooth sliceness Counterexample
• Ribbon disks and ribbon knots Definition
• A knot and its reverse mirror sum to a slice knot Example
• Tau of the unknot and trefoils from filtered staircases Example
• The square knot has a nontrivial character vanishing on a linking metabolizer Example
• Unknot, trefoils, and the 2-twisted positive Whitehead double Example
• A genus-g knot cobordism bounds the tau difference Lemma
• A slice disk induces a metabolizer on the branched-cover linking form Lemma
• A slice-branched-cover gluing produces a closed definite four-manifold Lemma
• Casson–Gordon signature defects vanish on the slice metabolizer Lemma
• Local resolution of ribbon singularities Lemma
• The auxiliary nu-minus threshold is bounded by smooth slice genus Lemma
• The CDR pattern zero-surgery homology cobordism Lemma
• The Hedden–Ording t=2 positive Whitehead double has tau zero and s two Lemma
• Weak injectivity and auxiliary-pattern reduction Lemma
• Smooth satellite filtrations with topologically slice images Open problem
• The slice-ribbon conjecture Open problem
• Reported refined s-invariant computations Remark
• Casson–Gordon signature-defect obstruction to sliceness Theorem
• Epsilon and Upsilon concordance data Theorem
• Every ribbon knot is smoothly slice Theorem
• Rasmussen's s-invariant bounds smooth four-ball genus Theorem
• Tau is an additive smooth concordance invariant and bounds four-ball genus Theorem
• Upsilon additivity, mirror symmetry, and four-ball genus bound Theorem

Dependency tree · two levels

20 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

• Charles Livingston, A Survey of Classical Knot Concordance, Handbook of Knot Theory, Chapter 7 (standard reference, not scraped)
• Arunima Ray, Slice Knots and Knot Concordance, Winter Braids Lecture Notes 8 (standard reference, not scraped)