Fox–Milnor factorization and slice obstructions — Examples and Counterexamples

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

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 $P=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$; direct multiplication gives $P^{T}VP=V^T$. 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.

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 $W=\mathrm{diag}(V,-V^T)$.

Let $P=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$. Direct multiplication gives $P^T{V}^{T}P=V$. The graph $M=\{(x,Px):x\in {\mathbb{Z}}^2\}$ is spanned by $e_1+e_4$ and $e_2+e_3$; projection onto the first factor identifies it with ${\mathbb{Z}}^2$, so it is a rank-$2$ direct summand. For any $x,y\in {\mathbb{Z}}^2$, $(x,Px{)}^{T}W(y,Py)=x^T(V-P^T{V}^{T}P)y=0$. Thus $M$ is a metabolizer.

The two blocks give $\det (V-t{V}^T)=(1-t{)}^2+t=t^2-t+1$ and $\det (-V^T+tV)=(t-1{)}^2+t=t^2-t+1$. 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 ${\mathrm{\Delta }}_S$.

By [F1] and [F2], ${\mathrm{\Delta }}_T(t)\doteq \det (V-t{V}^T)=(1-t{)}^2+t=t^2-t+1$. Thus $\mathrm{\mid }{\mathrm{\Delta }}_T(-1)\mathrm{\mid }=3$, since every Laurent unit $\pm t^k$ evaluates to $\pm 1$ at $-1$.

If ${\mathrm{\Delta }}_T(t)\doteq f(t)f(t^{-1})$ for some $f\in \mathbb{Z}[t^{\pm 1}]$, then evaluating at $t=-1$ gives $3=f(-1{)}^2$ or $-3=f(-1{)}^2$. Neither is possible because an integer square is nonnegative and $3$ is not an integer square. Hence the trefoil polynomial is not an integral reciprocal norm.

At $\omega =-1$, one has $H_{-1}(V)=2(V+V^T)=\left(\begin{array}{cc}-4& -2\\ -2& -4\end{array}\right)$, whose eigenvalues are $-2$ and $-6$. Since ${\mathrm{\Delta }}_T(-1)\mathrm{\ne }0$, its Levine–Tristram signature is $-2$. By [F4], a metabolic Seifert form would have signature $0$ there, so the trefoil Seifert form is not metabolic.

By [F6] and step2.1, the trefoil does not bound a locally flat properly embedded disk in the standard four-ball. Independently, if such a disk existed then [F5] would give a metabolic Seifert form, contradicting the signature calculation in step2.2 and [F4]. Thus the original locally flat sliceness claim is refuted by both necessary tests, with their TOP geometric inputs supplied rather than inferred from the numbers alone.