Does John’s reference to Christ as the Logos — the Word — rely on the Genesis creation narrative?

I think there is a connection other than just the word "logos" which itself is not directly equivalent to God;s sayings in the creation. Jesus is specifically linked to and made the instrumentality of the creation with the logos concept: "all things were made by him and without him was not any thing made that was made" and that certainly does ‘rely’ on Genesis 1 while at the same time reinterpreting the act of creation.

The other remarkable coincidence is the use of "light" and "life" and again, John freely reinterprets in a Targumic style both contemporising and effectively using the logos concept (whatever it is) to organically link Christ back to before the beginning of Genesis 1. There certainly are strong and deliberate echoes of Genesis in John’s Prologue and I think that a lot of the depth of meaning that is there in these few verses comes from the linkages with the creation account.

Live to serve : Serve to live