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The square-knot Seifert form is metabolic

Example

Assume full AC. Let $T$ be the right-handed trefoil obtained by plumbing two positive Hopf bands, with positive-push Seifert matrix $V = (\begin{array}{cc} - 1 & 0 \\ - 1 & - 1 \end{array})$ . Let $T^{†}$ be its reverse mirror and set $S = T # T^{†}$ . Then $S$ bounds a smooth proper disk in $B^{4}$ for which the squared radial function has only index- $0$ and index- $1$ interior critical points. A Seifert matrix for $S$ is $W = diag (V, - V^{T})$ , the rank- $2$ direct summand $M = ⟨ e_{1} + e_{4}, e_{2} + e_{3} ⟩ \subset Z^{4}$ is a metabolizer for $W$ , and $Δ_{S} (t) ≐ (t^{2} - t + 1)^{2}$ . In particular, with $f (t) = t^{2} - t + 1$ , the determinant is a Laurent-unit multiple of $f (t) f (t^{- 1})$ .

Example references

The positive-push matrix of the right-handed trefoil

The reverse-mirror convention

The fixed oriented connected-sum convention

The explicit ribbon disk for a reverse-mirror sum

The determinant normalization of $Δ_{S}$

Facts & Assumptions

Given: Full AC, the positive Hopf-plumbed trefoil $T$ , its reverse mirror $T^{†}$ , and the connected sum $S = T # T^{†}$ .

[A1]

Full AC is the family-of-nonempty-sets choice axiom used by the Seifert-pairing convention. The Axiom of Choice

[F1]

Full AC implies the Axiom of Countable Choice required by the ribbon-disk supplier. AC implies DC implies countable choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F2]

In the stated positive-push basis, the trefoil matrix is $V = (\begin{array}{cc} - 1 & 0 \\ - 1 & - 1 \end{array})$ . A positive Hopf plumbing gives the trefoil Seifert matrix

[F3]

The reverse mirror is obtained by the fixed orientation-reversing reflection and reversal of the knot orientation. Reverse mirror of an oriented knot

[F4]

The connected sum uses the fixed oriented ball-gluing convention, independent of the allowed local choices. Oriented connected sum and prime knot, The oriented connected sum is independent of its local choices

[F5]

Every smooth knot $K$ has a smooth proper disk for $K # K^{†}$ with only index- $0$ and index- $1$ interior radial critical points, under ${A C}_{ω}$ . A knot summed with its reverse mirror bounds a ribbon disk

[F6]

The Alexander polynomial satisfies $Δ_{S} (t) ≐ \det (W - t W^{T})$ for this positive-push matrix. The Seifert determinant is the knot Alexander polynomial

Verification

1.1A1F1F5givenconstruct

By [F1], the stated full-AC hypothesis supplies ${A C}_{ω}$ . Apply [F5] to $T$ ; its conclusion gives the required smooth proper disk for $S = T # T^{†}$ and the stated radial critical-point indices.

1.2F2F3algebra

The reverse-mirror surface is $- r (F)$ , so its positive normal is the reflected original positive normal: the ambient reflection and surface reversal cancel in the normal orientation. Thus the transported basis has matrix $- V$ , since reflection negates linking. For this particular trefoil matrix put $P = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ ; direct multiplication gives $P^{T} V P = V^{T}$ . Rechoose the basis of the reverse-mirror surface by $P$ . Its matrix is then $- V^{T}$ . This is a specified trefoil basis change, not a general image-basis formula for reverse mirrors.

2.1F2F4step 1.2construct

Form the boundary-connected-sum Seifert surface in the two summand balls. Cycles from different summands have zero linking because each bounds a $2$ -chain in its own ball, disjoint from the other summand. The combined basis therefore has matrix $W = diag (V, - V^{T})$ .

3.1F2step 2.1algebra

Let $P = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ . Direct multiplication gives $P^{T} V^{T} P = V$ . The graph $M = {(x, P x) : x \in Z^{2}}$ is spanned by $e_{1} + e_{4}$ and $e_{2} + e_{3}$ ; projection onto the first factor identifies it with $Z^{2}$ , so it is a rank- $2$ direct summand. For any $x, y \in Z^{2}$ , $(x, P x)^{T} W (y, P y) = x^{T} (V - P^{T} V^{T} P) y = 0$ . Thus $M$ is a metabolizer.

4.1F2F6step 2.1algebra∎

The two blocks give $\det (V - t V^{T}) = (1 - t)^{2} + t = t^{2} - t + 1$ and $\det (- V^{T} + t V) = (t - 1)^{2} + t = t^{2} - t + 1$ . Hence $\det (W - t W^{T}) = (t^{2} - t + 1)^{2}$ . For $f (t) = t^{2} - t + 1$ , one has $f (t^{- 1}) = t^{- 2} f (t)$ , so $\det (W - t W^{T}) = t^{2} f (t) f (t^{- 1})$ . By [F6], this is the claimed Fox–Milnor norm factorization of $Δ_{S}$ .

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Sources

Dale Rolfsen, Knots and Links (standard reference, not scraped)
Arunima Ray, Slice Knots and Knot Concordance (standard reference, not scraped)
Charles Livingston, A Survey of Classical Knot Concordance (standard reference, not scraped)