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The knot Alexander polynomial multiplies under connected sum

Statement

Assume the full Axiom of Choice. Let $K, J \subset S^{3}$ be smooth oriented knots, and let $K # J$ be their oriented connected sum for a connected-sum datum using the fixed gluing convention. Then

$Δ_{K # J} (t) ≐ Δ_{K} (t) Δ_{J} (t),$

where each $Δ$ is the generator of the zeroth elementary ideal of the one-variable Alexander module and $≐$ means equality up to multiplication by a unit $\pm t^{k}$ of $Z [t^{\pm 1}]$ . The statement is independent of the local connected-sum choices and of the Seifert surfaces and bases used to compute the polynomials.

Statement references

The oriented connected-sum construction and fixed gluing convention

Independence of local connected-sum choices

The positive-push Seifert matrix convention

The Seifert determinant formula

The polynomial and unit conventions

Facts & Assumptions

Given: Full AC, two smooth oriented knots $K, J \subset S^{3}$ , and a connected-sum datum as in the Definition.

[A1]

Full AC is the choice-function axiom used by the Alexander-module and knot-surface conventions. (The Axiom of Choice)

[F1]

The connected sum is formed by cutting each knot in a trivial-arc ball and gluing along the fixed orientation-matching boundary map. (Oriented connected sum and prime knot)

[F2]

For sufficiently small coordinate balls, changing the balls, subarcs, positive local frames, or oriented knot representatives does not change the oriented ambient-isotopy class of the connected sum. (The oriented connected sum is independent of its local choices)

[F3]

Full AC implies DC and hence the Axiom of Countable Choice. (AC implies DC implies countable choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F4]

A Seifert surface is a connected compact smooth oriented embedded surface with the given knot as its oriented boundary; under AC $_{ω}$ , every smooth oriented knot bounds one. (Oriented Seifert surfaces in the three-sphere, Every smooth oriented knot in $S^{3}$ bounds a Seifert surface)

[F5]

A connected compact knot surface has free fundamental group of rank $2 g (F)$ , and the first Hurewicz map identifies its first homology with the abelianization. Thus an ordered integral homology basis exists for each summand surface. (Finite surface normal forms, Jordan disks, and torsion control, The first Hurewicz map is abelianization)

[F6]

The Seifert matrix uses $V_{i j} = lk (a_{i}^{+}, a_{j})$ with the positive normal. (Seifert pairing and matrix of an oriented knot)

[F7]

The one-variable Alexander module is the positive-meridian module, $Δ_{K}$ is its zeroth-elementary-ideal generator, and for each connected knot Seifert surface and basis the module is presented by $V - t V^{T}$ , whose determinant generates that ideal up to a unit. (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)

[F8]

Ambient isotopies preserve the Alexander polynomial up to a unit, and transported/basis-changed Seifert matrices have the stated congruence relation. (The Alexander polynomial is an oriented link invariant, Seifert matrices of ambient-isotopic pairs are S-equivalent)

[F9]

A standard closed 3-ball is contractible, so its first singular homology is that of a point and vanishes. The singular chain complex of a point has $\partial_{1} = 0$ and $\partial_{2} = 1$ , hence $H_{1} (*; Z) = 0$ . (Contractible nonempty spaces have the homology of a point)

[F10]

Under AC $_{ω}$ , every smooth manifold with boundary, including each summand Seifert surface, has a smooth collar of its boundary. (Collar neighborhood theorem, Smooth collars of a manifold boundary)

[F11]

Under the supplier's AC $_{ω}$ hypothesis, its finite-graph-spine argument constructs a finite graph spine whose thickening is each connected compact surface with nonempty boundary; full AC supplies AC $_{ω}$ by [F3]. Thus $F$ and $G$ are thickenings of finite graphs. (Finite surface normal forms, Jordan disks, and torsion control, finite-graph-spine argument)

[F12]

Under AC $_{ω}$ , a smooth time-dependent vector field whose supports lie in a fixed compact set has a unique global evolution by diffeomorphisms on a compact time interval. In particular this applies to a smooth vector field on $S^{3}$ . (A compactly supported time-dependent field has a global time-one flow)

Proof

1.1A1F3F4F5givenchoose

By [A1] and [F3], the AC $_{ω}$ hypothesis of [F4] holds. Apply [F4] to choose connected Seifert surfaces $F$ for $K$ and $G$ for $J$ . By [F5], choose ordered integral bases for $H_{1} (F; Z)$ and $H_{1} (G; Z)$ .

2.1F1F2F3F12step 1.1construct

Reduce each given trivial-arc ball to a small coordinate ball as follows. Write its oriented ball-arc parametrization as $e : D^{3} \to B$ , with diameter on the first coordinate axis. On $B$ prescribe the inward field $X (e (x)) = D e_{x} (- x)$ . Extend it smoothly to $S^{3}$ , tangent to the whole knot: near each arc endpoint take coordinates $(s, u, v)$ with the knot $u = v = 0$ and the ball boundary $s = 0$ . The two normal components on the ball side vanish on the axis, so each is $u a + v b$ with smooth coefficients, by integrating normal derivatives. Extend these coefficients and the tangential component across $s = 0$ ; away from the endpoints extend across the sphere in ordinary boundary charts. A finite partition of unity patches these extensions, all agreeing with $X$ on $B$ , and a cutoff makes the field zero away from a neighborhood of $B$ while preserving tangency to the knot. This is the tangent-field extension construction in the proof of [F2], now applied at the boundary of the given ball. By [F12] its flow $H_{t}$ exists for every fixed finite time interval; uniqueness gives $H_{t} (e (x)) = e (e^{- t} x)$ for $x \in D^{3}$ . The flow preserves the oriented knot setwise, and for large finite $T$ its image of $B$ is an arbitrarily small coordinate ball in the interior chart $e$ . Transport the boundary parametrization as $H_{t} \circ e$ throughout. The fixed gluing map in these parametrizations is unchanged, so the flows identify the cut-and-glued pairs and their boundary collars throughout the isotopy, just as in the representative-transport argument in the proof of [F2]. Thus the original connected sum is ambient isotopic to one using small coordinate balls.

3.1F1F2F5F10F11step 1.1step 2.1givenconstruct

By step 2.1 and [F2], it suffices to use standard round coordinate balls whose trivial-arc neighborhoods lie in boundary collars of $F$ and $G$ ; such collars exist by [F10]. Their complementary pieces are standard closed 3-balls. Remove a small boundary half-disk from each surface and join the exposed boundary arcs by the standard band in the gluing collar. The resulting embedded oriented surface $S$ has boundary $K # J$ . By [F11], each summand surface is a thickening of a finite graph. Include a path from each attaching arc to its graph as a tree edge, then retract the band to its core interval. The two graphs are joined by one further bridge, so no cycle is added. Their old cycle bases therefore form a basis of $H_{1} (S; Z)$ .

4.1F1F6F9step 3.1givenconstruct

Use the union basis from step 3.1. Each basis cycle and its positive push-off can be kept in the interior of its own complementary 3-ball. By [F9], a cycle in one ball bounds an integral $2$ -chain there; that chain is disjoint from every cycle in the other ball. Thus both cross-linking entries between the two summands are zero. The within-summand entries are unchanged, so the Seifert matrix of $S$ is $V_{K} \oplus V_{J}$ .

5.1F1F2F7F8step 2.1step 4.1algebra∎

The determinant of the block matrix satisfies $\det ((V_{K} \oplus V_{J}) - t (V_{K} \oplus V_{J})^{T}) = \det (V_{K} - t V_{K}^{T}) \det (V_{J} - t V_{J}^{T})$ . Apply [F7] to $S, F, G$ to identify these three determinants with generators of their respective $E_{0}$ ideals, up to units. By step 2.1 and [F1], [F2], and [F8] this identity gives the formula for the specified connected sum and is unchanged by the datum, surface, basis, or representative choices.

Diagram

This schematic shows the two summand surfaces in disjoint complementary 3-balls and the boundary band used to form $S$ . The cycles $a_{i}$ and $b_{j}$ remain in separate balls; the proof uses the separating balls to make the cross-linking terms zero. The drawing records the construction and is not a proof premise.

\begin{tikzpicture}[scale=0.9]
  \draw[fill=blue!8,thick,rounded corners] (-3,-1) rectangle (-0.5,1);
  \draw[fill=orange!10,thick,rounded corners] (0.5,-1) rectangle (3,1);
  \node at (-1.75,0.75) {$F$};
  \node at (1.75,0.75) {$G$};
  \draw[thick,blue] (-2.4,0) ellipse (0.55 and 0.35);
  \draw[thick,orange] (1.4,0) ellipse (0.55 and 0.35);
  \node[blue] at (-2.4,0) {$a_i$};
  \node[orange] at (1.4,0) {$b_j$};
  \draw[fill=gray!20,thick] (-0.5,-0.25) rectangle (0.5,0.25);
  \node at (0,0) {band};
  \node[below] at (-1.75,-1.2) {complementary 3-ball};
  \node[below] at (1.75,-1.2) {complementary 3-ball};
\end{tikzpicture}

Depends on

Used by

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116 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)