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Seifert pairing and matrix of an oriented knot

Definition

Assume the Axiom of Choice. Let $K : S^{1} ↪ S^{3}$ be a smooth oriented knot in the standard oriented three-sphere, and let $F \subset S^{3}$ be a connected compact smooth oriented embedded surface with boundary $K$ , oriented so that its boundary orientation is the given orientation of $K$ . Let $g$ be the nonnegative integer determined by $χ (F) = 1 - 2 g$ . Choose an ordered integral basis $a_{1}, \dots, a_{2 g}$ of the first singular homology group $H_{1} (F; Z)$ , represented by integral one-cycles on $F$ . Orient the normal line of $F$ by requiring $(n, u, v)$ to be positively oriented in $S^{3}$ whenever $(u, v)$ is a positive tangent basis of $F$ . The positive push-off $a_{i}^{+}$ is the cycle carried to a sufficiently small positive normal level in a bicollar of $F$ .

For disjoint oriented closed curves $J, L \subset S^{3}$ , orient a meridian $μ_{L}$ of $L$ so that a positive tangent to $L$ followed by a positive basis of its normal disk gives the ambient orientation. The linking number $lk (J, L)$ is the integer determined by $[J] = lk (J, L) [μ_{L}]$ in $H_{1} (S^{3} ∖ L; Z)$ , using the Alexander-duality identification that sends $[μ_{L}]$ to $1$ . Extend linking additively over integral one-cycles. The Seifert pairing is $θ_{F} (x, y) = lk (x^{+}, y)$ , and its matrix in the chosen basis is $V = (V_{i j})_{i, j = 1}^{2 g}$ , where $V_{i j} = θ_{F} (a_{i}, a_{j}) = lk (a_{i}^{+}, a_{j})$ . The surface and basis are part of the data; this definition asserts no knot-only matrix invariant.

Definition references

A Seifert surface is connected, compact, smooth, oriented, embedded, and has the oriented knot as its boundary; its genus satisfies $χ (F) = 1 - 2 g (F)$ .

A genus- $g$ surface with one boundary component has $H_{1} (F; Z) ≅ Z^{2 g}$ and the standard handle normal form.

For one component, Alexander duality normalizes the positive meridian as the generator of $H_{1} (S^{3} ∖ K; Z)$ .

The topological linking number is the positive-meridian coefficient and agrees with the signed crossing and surface-intersection conventions.

Full Axiom of Choice used by the Alexander-duality linking convention.

Full choice implies the countable-choice hypothesis of the surface normal-form suppliers.

The linking formula descends to a bilinear pairing on homology classes and is independent of the small positive push-off. ↗

Depends on

Used by

Knot Alexander normalization, symmetry, and genus bound Corollary
The knot Alexander polynomial multiplies under connected sum Corollary
A higher cyclic branched cover can have infinite first homology Counterexample
An abstract annulus need not be a smooth concordance Counterexample
The knot-normalized Alexander fraction need not be a Laurent polynomial Counterexample
The square knot is slice despite its nonunit Alexander polynomial Counterexample
The trefoil fails both algebraic sliceness tests Counterexample
The unknot has a zero-order Seifert matrix Counterexample
Unimodular skew part does not force metabolicity Counterexample
Integral Witt classes and rational Seifert isometric structures Definition
Knot Arf invariant from the Seifert form Definition
Levine–Tristram Hermitian form and signature Definition
Metabolic Seifert forms and algebraic knot concordance Definition
S-equivalence and Conway normalization Definition
S-equivalence of integral Seifert forms Definition
The square knot has a nontrivial character vanishing on a linking metabolizer Example
The square-knot Seifert form is metabolic Example
Trefoil and figure-eight Alexander polynomials from Seifert matrices Example
Trefoil and figure-eight Levine–Tristram and Arf calculations Example
Trefoil Seifert matrix and basis changes Example
Unknot, trefoil, and positive untwisted Whitehead double cases Example
Worked cases for metabolic Seifert forms Example
Worked cases: double branched covers of the trefoil, figure-eight knot, and unknot Example
A positive Hopf plumbing gives the trefoil Seifert matrix Lemma
A slice disk induces a Seifert metabolizer Lemma
An Alexander-one knot supplies good-group disk-embedding data Lemma
An Alexander-one Seifert form is S-equivalent to the empty form Lemma
An invertible Seifert matrix gives a rational isometric structure Lemma
Every integral Seifert form is realized by a knot Lemma
Homology of the double branched cover from a Seifert matrix Lemma
One crossing change changes a Seifert matrix by rank at most one Lemma
One-sided Levine–Tristram signature limits exist Lemma
Root-product test for cyclic branched-cover homology Lemma
Seifert block presentation of cyclic branched-cover homology Lemma
Seifert matrices of a knot are S-equivalent Lemma
Seifert matrices of ambient-isotopic pairs are S-equivalent Lemma
Seifert stabilization preserves the Levine–Tristram signature Lemma
Seifert-matrix certificate for the Conway diagram Lemma
Surgery presentation from a Seifert surface Lemma
The cut-open cover presentation of the Alexander module Lemma

…and 16 more results.

Dependency tree · two levels

75 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

Dale Rolfsen, Knots and Links, Chapter 8.A (standard reference, not scraped)
Alessia Cattabriga and Stefano Francaviglia, Knot Theory and Applications: Lecture Notes, Section 6.2.3 (standard reference, not scraped)