I saw your post, although old, I thought I would respond. Jesse Knapp would be my Great Grandfather. I am relatively new to genealogy. My dad started digging into it about 10 years ago. Well I thought I would start digging in as well. So tonight I believe I may have traced the Knapp line back across the pond to Bures,UK. My dad had found records taking us back via John Reuben Knapp to Cyrus C. Knapp. I was able to confirm his findings separately via census records. The records seem to be a bit thin due to the westward expansion. It didn't help that John Reuben Knapp used either name and moved frequently, traveling from Ohio to Kansas to Colorado, Nebraska. Luckily he had enough kids that there were usually a kid or two along with his wife and birthday to conclude it was him. I found this site helpful https://familysearch.org/ . From there he(John R[e]uben) married a younger wife and later moved to Beauregard, Louisiana. Census showed John R. Knapp's Parents were from NY and PENN and that matched up with Cyrus C. Knapp. From there the trail got a bit cold. All I could find was that his parent were from Dansbury, CT. I then stumbled across this site http://freepages.genealogy.rootsweb.ancestry.com/~knappdb/nicholas/p000001n.htm#I26935http://freepages.genealogy.rootsweb.ancestry.com/~knappdb/nicholas/p000001n.htm#I26935 that went from John Reuben all the way back to 1636. Now I have not confirmed this data but it still is a bit exciting to me.
I have about 100 of the relative and their info in a family tree map on Geni.com and I am mapping the census data on Goolge maps. I might be able to provide that if your interested once I have time to clean it up.
Anyway, hope this helps. Let me know if you have any sources to confirm any of this.
If you wanted more of the lower branch my dad knows many of his 80+ first cousins,uncles, and aunts.