Knox was tied with Hunter Mahan at 6-under par after a perfect day for golf along the Pacific Ocean.
Three courses are used during the first three rounds of the tournament, but they do not all have the same par.
Knox shot a 6-under 64 at the par-70 Monterey Peninsula course while Mahan turned in a 6-under 66 at the par-72 Pebble Beach Golf Links.
Brandt Snedeker was among those at 4-under. Snedeker finished second in each of the last two PGA Tour events -- trailing Tiger Woods by four shots two weeks ago and Phil Mickelson by four shots last week.
Knox, 27, is a native of Scotland and a current resident of Jacksonville, Fla. He has only limited status on the tour and needs to take advantage of his few opportunities to compete. Knox missed the cut at the Sony Open in his only other appearance on the tour this year.
Mahan won twice in 2012, one of his victories coming in the Match Play Championship.
Mickelson, who shot a 60 en route to his win last week at the Phoenix Open, settled for a 1-under 69 at Monterey Peninsula on Thursday. He is also the defending champion of this event.
Swim Week Miami Beach 2014 [PHOTOS]