Oriented Seifert surfaces in the three-sphere

Depends on

• Oriented links in the three-sphere and ambient isotopy
• Finite surface normal forms, Jordan disks, and torsion control
• The Axiom of Countable Choice ($\mathrm{AC}_\omega$)

Used by

• Knot Alexander normalization, symmetry, and genus bound Corollary
• The knot Alexander polynomial multiplies under connected sum Corollary
• A genus-one spanning surface for the unknot is not minimal Counterexample
• A higher cyclic branched cover can have infinite first homology Counterexample
• The knot-normalized Alexander fraction need not be a Laurent polynomial Counterexample
• The unknot has a zero-order Seifert matrix Counterexample
• Seifert pairing and matrix of an oriented knot Definition
• Seifert surfaces and smooth knot genus Definition
• The Alexander filtration and tau level Definition
• Tau of the unknot and trefoils from filtered staircases Example
• Trefoil and figure-eight Alexander polynomials from Seifert matrices Example
• Worked cases: double branched covers of the trefoil, figure-eight knot, and unknot Example
• Worked Seifert-algorithm counts for the trefoil, figure-eight, and kinked unknot Example
• A positive Hopf-plumbed surface for a torus knot Lemma
• An Alexander-one knot has a trivial Alexander basis Lemma
• Homology of the double branched cover from a Seifert matrix Lemma
• Root-product test for cyclic branched-cover homology Lemma
• Seifert block presentation of cyclic branched-cover homology Lemma
• Seifert disks and bands form an embedded oriented spanning surface Lemma
• Seifert matrices of a knot are S-equivalent Lemma
• Seifert matrices of ambient-isotopic pairs are S-equivalent Lemma
• Surgery presentation from a Seifert surface Lemma
• The cut-open cover presentation of the Alexander module Lemma
• The filtered knot Floer complex of a connected sum is a tensor product Lemma
• The skew part of a Seifert matrix is the surface intersection form Lemma
• The square Seifert presentation generates the knot Alexander polynomial Lemma
• Well-definedness of the Seifert pairing Lemma
• A Seifert framing has zero self-linking Theorem
• A Seifert matrix presents the one-variable Alexander module Theorem
• Cyclic-cover homology order from the Alexander polynomial Theorem
• Euler characteristic and genus of a Seifert disk-and-band surface Theorem
• Knot determinant is the order of double-cover homology Theorem
• The Seifert determinant is the knot Alexander polynomial Theorem

Dependency tree · two levels

35 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

• Danny Calegari, Introduction to Knot Theory, §3.1, pp.21–22 (standard reference, not scraped)
• Alessia Cattabriga and Stefano Francaviglia, Knot Theory and Applications: Lecture Notes, §4.3, pp.21–23 (standard reference, not scraped)