We can represent every element of the group $K^{-1}(X)=\tilde{K}(SX)$ by a isomorphism of trivial vector bundles $L:\, X\times \mathbb{C}^k \to X\times \mathbb{C}^k$ because $SX$ is the union of two cones over $X$ with intersection $X$.

We also have two products $K^{-1}(X)\otimes K^{-1}(X) \to K(X)$ and $K(X)\otimes K^{-1}(X) \to K^{-1}(X)$ which are given by the ring structure in $K^*(X)=K(X)\oplus K^{-1}(X)$.

My questions are: can we define explicitly these products using the representation above? Is there some geometric interpretation of these maps?