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Trefoil and figure-eight Alexander polynomials from Seifert matrices

Example

Assume the full Axiom of Choice. Let $T$ and $E$ be the oriented boundaries of the standard one-boundary square plumbings of two positive Hopf bands and of one positive with one negative Hopf band, respectively; these are the right-handed trefoil and the figure-eight knot. A positive Hopf band is an oriented annulus with unknotted core whose induced boundary components have linking number $+ 1$ ; its core, the central simple closed curve, has self-pairing $- 1$ . A negative Hopf band also has unknotted core and has boundary linking number $- 1$ and core self-pairing $+ 1$ . A square plumbing identifies a square in the boundary collars of the two annuli so that its edges alternate between them, then smooths the corners; use the gluing for which the resulting boundary is one component. Order the two cores $a, b$ with the positive Hopf core first, orient them so $a \cdot_{F} b = + 1$ , and use the standard plumbing side for which $a^{+}$ bounds a disk disjoint from $b$ . In the positive-push convention $V_{i j} = lk (a_{i}^{+}, a_{j})$ , their Seifert matrices are

$V_{T} = (\begin{array}{cc} - 1 & 0 \\ - 1 & - 1 \end{array}), V_{E} = (\begin{array}{cc} - 1 & 0 \\ - 1 & 1 \end{array}) .$

Their determinant representatives are $Δ_{T} (t) = 1 - t + t^{2}$ and $Δ_{E} (t) = - 1 + 3 t - t^{2}$ . They both evaluate to $1$ at $t = 1$ ; at $t = - 1$ the displayed representatives evaluate to $3$ and $- 5$ , so the unit-invariant absolute values are $3$ and $5$ . These determinant classes are unchanged up to units by oriented ambient isotopy and integral basis changes. The connected sum satisfies

$Δ_{T # E} (t) ≐ (1 - t + t^{2}) (- 1 + 3 t - t^{2}) = - t^{4} + 4 t^{3} - 5 t^{2} + 4 t - 1.$

This displayed representative has values $1$ at $t = 1$ and $- 15$ at $t = - 1$ .

Example references

Positive-push Seifert matrix entries

Connected compact oriented spanning surfaces and the genus formula

The positive-meridian Alexander module

The polynomial generator and unit convention

The oriented connected-sum operation

The skew part records the oriented surface intersection

The determinant formula

Evaluation, reciprocity, and Laurent width

Connected-sum multiplicativity

Knot Theory Lecture Notes, Chapter 7, Examples 7.4–7.5; alternate ordered-core sign convention

Facts & Assumptions

Given: Full AC and the specified positive-positive and positive-negative Hopf-band plumbings with one boundary component.

[A1]

Full AC is the choice-function axiom assumed by the Alexander-module and Seifert-matrix definitions. (The Axiom of Choice)

[F1]

The positive-push matrix is $V_{i j} = lk (a_{i}^{+}, a_{j})$ . (Seifert pairing and matrix of an oriented knot)

[F2]

For every ordered integral homology basis, $V - V^{T} = J_{F}$ , and the surface intersection matrix $J_{F}$ is unimodular. (The skew part of a Seifert matrix is the surface intersection form)

[F3]

A Seifert surface is connected, compact, smooth, oriented and embedded with its given knot as oriented boundary; a connected compact surface with one boundary has free fundamental group of rank $2 g (F)$ , and its first homology is the free abelianization. For genus one, this gives a free rank-two group. (Oriented Seifert surfaces in the three-sphere, Finite surface normal forms, Jordan disks, and torsion control, The first Hurewicz map is abelianization)

[F4]

The positive-meridian module is presented by $V - t V^{T}$ , its $E_{0}$ ideal is generated by the determinant, and that determinant is associated to the polynomial generator. (The one-variable Alexander module of an oriented link, The Alexander polynomial from the zeroth elementary ideal, A Seifert matrix presents the one-variable Alexander module, The square Seifert presentation generates the knot Alexander polynomial, The Seifert determinant is the knot Alexander polynomial)

[F5]

The exponent-width convention gives $Δ_{K} (1) = \pm 1$ and reciprocal symmetry up to units. (Knot Alexander normalization, symmetry, and genus bound)

[F6]

For oriented knots $K, J$ , $Δ_{K # J} ≐ Δ_{K} Δ_{J}$ . (The knot Alexander polynomial multiplies under connected sum)

[F7]

The connected sum uses the fixed oriented gluing convention and represents a one-component knot. (Oriented connected sum and prime knot)

[F8]

Full AC implies the Axiom of Countable Choice, the surface-normal-form hypothesis. (AC implies DC implies countable choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F9]

Ambient isotopy transports a Seifert matrix by unimodular congruence and preserves the Alexander polynomial up to a unit. (Seifert matrices of ambient-isotopic pairs are S-equivalent, The Alexander polynomial is an oriented link invariant)

Verification

1.1A1F2F3F8givenalgebra

Each plumbing has Euler characteristic $0 + 0 - 1 = - 1$ , using a cell structure respecting the common square, since each annulus has Euler characteristic $0$ and the square has Euler characteristic $1$ . Its boundary is one circle by the stipulated gluing, so $χ = 1 - 2 g$ gives genus one. By [F3], $H_{1} (F; Z)$ is free of rank two. Choose any ordered basis $u, v$ ; [F2] makes its intersection matrix $J_{0}$ skew-symmetric and unimodular, hence $\det J_{0} = 1$ . Write the two core classes $a, b$ in this basis as the columns of an integer matrix $P$ . The plumbing gives $a \cdot_{F} b = 1$ , so their intersection matrix is $J = (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array}) = P^{T} J_{0} P$ . Taking determinants gives $1 = (\det P)^{2}$ , hence $\det P = \pm 1$ ; therefore $a, b$ is an integral basis.

2.1A1F1F2step 1.1givenalgebra

Equally oriented parallel cores of an annulus transport to its two induced boundary orientations with one reversed, so their linking is minus the boundary linking. Returning a small positive normal push in the complement of the other parallel core shows that each Hopf core self-pairing is minus its boundary linking. Thus the Hopf-band signs give $V_{11} = V_{22} = - 1$ for $T$ and $V_{11} = - 1, V_{22} = 1$ for $E$ . The specified disk bounded by $a^{+}$ is disjoint from $b$ , so its class vanishes in the complement of $b$ and the linking definition gives $V_{12} = 0$ . Now that $a, b$ is a basis, [F2] yields $V_{12} - V_{21} = a \cdot_{F} b = 1$ , hence $V_{21} = - 1$ . This proves the displayed matrices.

3.1A1F4F5step 2.1algebra

For $V_{T}$ , the matrix $V_{T} - t V_{T}^{T}$ is $(\begin{array}{cc} t - 1 & t \\ - 1 & t - 1 \end{array})$ , with determinant $(1 - t)^{2} + t = 1 - t + t^{2}$ . For $V_{E}$ , the matrix $V_{E} - t V_{E}^{T}$ is $(\begin{array}{cc} t - 1 & t \\ - 1 & 1 - t \end{array})$ , with determinant $(t - 1) (1 - t) + t = - 1 + 3 t - t^{2}$ . By [F4], these determinants represent $Δ_{T}$ and $Δ_{E}$ up to units. Substitution gives the stated values at $1$ and $- 1$ ; every unit is $\pm t^{k}$ , so the absolute values at both substitutions are independent of the representative. Their values $1$ at $1$ agree with [F5].

4.1A1F6F7step 3.1algebra

By [F7], $T # E$ is the connected sum in the fixed oriented convention. Apply [F6] to obtain $Δ_{T # E} ≐ Δ_{T} Δ_{E}$ . Multiplying the two displayed quadratics gives $- t^{4} + 4 t^{3} - 5 t^{2} + 4 t - 1$ ; substituting $1$ and $- 1$ gives $1$ and $- 15$ .

5.1A1F9step 3.1algebra∎

By [F9], orientation-preserving ambient isotopy preserves the polynomial class, and an integral basis change is a unimodular congruence of the matrix. Its determinant changes by $(\det P)^{2} = 1$ , so the determinant class and the displayed unit-invariant evaluations are unchanged.

Diagram

The panels show the local square-plumbing charts: the horizontal and vertical core arcs cross transversely once in the shaded common square and continue outside the displayed neighborhoods to close. The labels $+, +$ and $+, -$ give the Hopf-band signs for the trefoil and figure-eight surfaces; the local charts do not depict their twists. Orient each square so the ordered rightward and upward tangents are positive; the arrows then give $a \cdot_{F} b = + 1$ . This schematic records the local construction; the proof computes the matrix entries in words.

\begin{tikzpicture}[scale=0.85]
  % Trefoil: positive horizontal and vertical Hopf bands.
  \draw[fill=blue!10,thick] (-4.5,-0.4) rectangle (-0.6,0.4);
  \draw[fill=orange!10,thick] (-2.9,-1.8) rectangle (-2.1,1.8);
  \fill[gray!40] (-2.9,-0.4) rectangle (-2.1,0.4);
  \draw[thick,->,blue] (-4.1,0) -- (-0.95,0);
  \draw[thick,->,orange] (-2.5,-1.45) -- (-2.5,1.45);
  \node at (-2.5,2.2) {$T:\ +,+$};
  \node[blue] at (-3.8,0.75) {$a$-band};
  \node[orange] at (-1.2,1.5) {$b$-band};
  \node at (-2.5,-2.2) {$a\cdot_F b=+1$};

  % Figure-eight: positive horizontal and negative vertical Hopf bands.
  \draw[fill=blue!10,thick] (0.6,-0.4) rectangle (4.5,0.4);
  \draw[fill=orange!10,thick] (2.1,-1.8) rectangle (2.9,1.8);
  \fill[gray!40] (2.1,-0.4) rectangle (2.9,0.4);
  \draw[thick,->,blue] (0.95,0) -- (4.1,0);
  \draw[thick,->,orange] (2.5,-1.45) -- (2.5,1.45);
  \node at (2.5,2.2) {$E:\ +,-$};
  \node[blue] at (1.2,0.75) {$a$-band};
  \node[orange] at (3.8,1.5) {$b$-band};
  \node at (2.5,-2.2) {$a\cdot_F b=+1$};
\end{tikzpicture}

Depends on

Used by

A primitive sixth root makes the trefoil Hermitian form singular Counterexample
Trefoil and figure-eight Levine–Tristram and Arf calculations Example

Dependency tree · two levels

108 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 (standard reference, not scraped)
Alessia Cattabriga and Stefano Francaviglia, Knot Theory and Applications: Lecture Notes (standard reference, not scraped)
Knot Theory Lecture Notes, Chapter 7: Seifert Matrices and Derived Invariants (standard reference, not scraped)