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Seifert matrices of ambient-isotopic pairs are S-equivalent

Statement

Assume the Axiom of Choice. Let $(K, F)$ and $(K^{'}, F^{'})$ be oriented knot and spanning-surface pairs in the oriented standard three-sphere, and let $h_{t} : S^{3} \to S^{3}$ be a smooth ambient isotopy with $h_{0}$ the identity, $h_{1} (K) = K^{'}$ , and $h_{1} (F) = F^{'}$ , preserving the given orientations. If $V$ and $V^{'}$ are their Seifert matrices in any integral bases, then the basis transported by $h_{1}$ has matrix $V$ , and any other target basis with transition matrix $P$ has matrix $V^{'} = P^{T} V P$ . Hence $V$ and $V^{'}$ are S-equivalent, meaning they are related by a finite sequence of integral unimodular congruences and the two elementary rank-two enlargement moves or their reverses.

Statement references

The positive-push Seifert pairing and matrix

The congruence and elementary enlargement moves defining S-equivalence

Well-definedness and ambient linking convention

Oriented knots and ambient isotopy

Facts & Assumptions

Given: Full AC, two oriented knot-surface pairs and an orientation-preserving smooth ambient isotopy carrying the first pair to the second.

[A1]

Full AC supplies the Alexander-duality linking convention used by the Seifert pairing. The Axiom of Choice

[F1]

The Seifert pairing is bilinear on homology classes and is computed by linking positive normal push-offs. Seifert pairing and matrix of an oriented knot, Well-definedness of the Seifert pairing

[F2]

An orientation-preserving ambient isotopy induces homology isomorphisms on complements and carries positive meridians to positive meridians; naturality of Alexander duality preserves the total-linking coefficient. Oriented links in the three-sphere and ambient isotopy, Singular chains and singular homology are covariantly functorial, The one-variable Alexander module of an oriented link

[F3]

The positive normal is fixed by the ambient and surface orientations, so an orientation-preserving diffeomorphism carrying the oriented surface carries positive normal levels to positive normal levels. Seifert pairing and matrix of an oriented knot

[F4]

S-equivalence is generated by integral unimodular congruences, the two elementary rank-two enlargements, and their reverses. S-equivalence of integral Seifert matrices

Proof

1.1A1F1F2F3givenconstruct

Put $h = h_{1}$ . Because $h$ preserves the ambient orientation and the orientations of $K, F$ , [F3] sends each positive normal push-off of a curve $a_{i}$ to a positive normal push-off of $h (a_{i})$ , up to a homotopy within the positive collar. By [F2] the induced map on the complement preserves the positive-meridian coefficient, so $lk (a_{i}^{+}, a_{j}) = lk (h (a_{i})^{+}, h (a_{j})) .$ Thus the transported basis $b_{i} = h_{*} [a_{i}]$ of $H_{1} (F^{'}; Z)$ has exactly the same matrix $V$ .

2.1F1F4step 1.1givenalgebra∎

Let $c_{1}, \dots, c_{n}$ be any chosen integral basis on $F^{'}$ . There is a unique integer matrix $P$ with $[c_{j}] = \sum_{i} P_{i j} b_{i}$ ; since both lists are integral bases, $P \in {G L}_{n} (Z)$ . Bilinearity [F1] gives the matrix in the $c$ basis as $V^{'} = P^{T} V P$ . By [F4], this unimodular congruence is one of the defining S-equivalence moves, so the two matrices are S-equivalent.

Remarks

The conclusion uses only the congruence move in the stated S-equivalence relation; the elementary enlargements are included to fix the relation's full convention for later consumers.

Depends on

Used by

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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)
Knot Theory Lecture Notes, Chapter 7: Seifert Matrices and Derived Invariants (standard reference, not scraped)