Odd K-theory of the sphere bundle of two tautological lines
= Odd K-theory of the sphere bundle of two tautological lines
{c}
{title2=$K^{-1}(S(\gamma\oplus\gamma))$}
For $E=\gamma\oplus\gamma$ over $\mathbb{CP}^n$ and $t=1-[\overline\gamma]$, the K-theory Euler class is $t^2$. Hence
$$
K^{-1}(S(E))
\cong\ker(t^2:\mathbb Z[t]/(t^{n+1})\to\mathbb Z[t]/(t^{n+1}))
=\mathbb Z\{t^{n-1},t^n\}
$$
for $n\geq1$.
For $n=0$, the base is a point, the sphere bundle is $S^3$, and its odd K-theory is $\mathbb Z$.