What is "the lap of the liver"?

I suppose you're not talking about the archaic word for the tissue surrounding the liver? ("A Dictionary of Archaic and Provincial Words" by John Orchard Halliwell)

If you mean the surgical procedure, here's an explanation

http://www.surgeryencyclopedia.com/La-Pa/Laparotom...

C/EBP- b/ LAP Controls Down-regulation of Albumin Gene Transcription during Liver Regeneration

http://www.jbc.org/cgi/reprint/271/36/22262.pdf

LAP is not an anatomical feature rather its an enzyme called Liver Activator Protein. Or, you could be referring to this:

"...Under the lap of my liver, where all the secrets of my heart lie"

A Chinese medical art called Qi Gong that believes the body betrays its trouble by the sound it makes. It requires that you sit quietly cross-legged with you hands in front of you forming a lap. This is referred to as the lap of the liver.