= Solution
Orient $S^3$ as the boundary of $B^4$. Push the interior of an oriented <Seifert surface> $F$ slightly into $B^4$, and let $X_F$ be the double <branched covering> of $B^4$ over this pushed-in surface. Its boundary is the <two-fold branched cover of a knot>. Define the \b[knot signature] by
$$
\boxed{\sigma(K)=\operatorname{sign}Q_{X_F}.}
$$
The <signature> of a possibly degenerate <intersection form> means the number of positive eigenvalues minus the number of negative eigenvalues; its radical contributes zero.
Here is why the definition is independent of $F$. Given two choices $F_0,F_1$, glue their branched covers along the common boundary to obtain
$$
X=X_{F_0}\cup_{\Sigma_2(K)}(-X_{F_1}).
$$
This is the double <branched covering> of $S^4$ over the closed oriented surface obtained by gluing the pushed-in $F_0$ to $-F_1$. Every closed oriented surface in $S^4$ admits a <Seifert hypersurface>: its complement has an integral meridional class, represent it by a map to the circle, and take a <regular value>. Near the surface choose the angular map in its trivial normal disk bundle. The closure of a suitable regular fiber is a compact oriented three-manifold bounding the surface. Its trivial normal framing can be chosen to match this construction.
Push the interior of this <Seifert hypersurface> into $B^5$. The double <branched covering> of $B^5$ over the resulting properly embedded three-manifold has boundary $X$. The existence of the cover follows from the meridian homomorphism to $\mathbb Z/2$; smoothness near the branch set is the local map $(z,u)\mapsto(z^2,u)$. By the hypothesis about boundaries of five-manifolds, $\operatorname{sign}X=0$. <Novikov additivity> for gluing along an entire closed three-dimensional boundary gives
$$
0=\operatorname{sign}X=\operatorname{sign}X_{F_0}-\operatorname{sign}X_{F_1}.
$$
Thus the <knot signature> is independent of the <Seifert surface>. An <isotopy> of the <knot> carries the construction to an equivalent one, so it is also a <knot invariant>.
To calculate its <intersection form>, express $F$ as a disk with $2g$ bands. The double <branched covering> of $B^4$ over the pushed-in disk is again $B^4$. Each band lifts to a two-handle. The capped lifted cores give a basis of $H_2(X_F;\mathbb Z)$ corresponding to the band-core basis of $H_1(F;\mathbb Z)$. The framing of the $i$th lifted attaching circle is $2\operatorname{lk}(a_i^+,a_i)=2V_{ii}$. For distinct bands, the two sheets contribute the two push-off linkings $V_{ij}$ and $V_{ji}$, so the mutual <linking number> is $V_{ij}+V_{ji}$. Hence
$$
Q_{X_F}=V+V^T,\qquad
\boxed{\sigma(K)=\operatorname{sign}(V+V^T).}
$$
A different integral basis changes this matrix by <matrix congruence> and leaves its <signature> unchanged.
In Figure 7 the upper knot is a <connected sum of knots> $T\mathbin{\#}T$, while the lower one is $T\mathbin{\#}\overline T$, where $T$ is the positive <trefoil knot>. Both <knot groups> have the \b[same presentation]
$$
\boxed{\langle a,b,c\mid aba=bab,\ aca=cac\rangle.}
$$
To see this, the <Seifert-van Kampen theorem> expresses the <knot group> of a <connected sum of knots> as the <amalgamated free product> of the two summand <knot groups>, amalgamating their <meridians of a knot>. A <trefoil knot> has presentation $\langle a,b\mid aba=bab\rangle$ with $a$ a meridian. Reflection gives the same abstract group for the mirror. If it reverses the chosen meridian, send both generators to their inverses; the inverse braid relation is again the same relation. Thus the meridian amalgamation yields the displayed <group presentation> in both cases.
A band basis for the positive <trefoil knot> gives
$$
V_T=\begin{pmatrix}-1&1\\0&-1\end{pmatrix},
$$
whose symmetrization is negative definite. Therefore $\sigma(T)=-2$ and $\sigma(\overline T)=2$. Boundary-connected-summing <Seifert surfaces> makes the <Seifert matrix> block diagonal, so the <knot signature> is additive under <connected sum of knots>. Consequently
$$
\boxed{\sigma(K_1)=-4,\qquad \sigma(K_2)=0.}
$$
The upper knot and lower knot are therefore not isotopic, even though their <knot groups> are isomorphic. With the opposite global <knot signature> convention the first value is $+4$, and the distinction is unchanged.
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