Burton, of Evans, was in second place after two rounds, but had 7-over-par 78 on Saturday and dropped into a tie for 13th place, five shots off the lead.
Knox, who was five back after 36 holes, is now two back after shooting 73.
He was able to pick up ground on David Noll Jr., who has led after each round. He still leads by one shot despite 76 on Saturday and is at 1-over 214 for 54 holes.
Fred Purcell, of LaGrange (69) and Travis Williamson, of Brunswick (72) are tied for second place.
Knox is tied with two others for fourth place.
Knox, a two-time Georgia Mid-Amateur champion, is seeking his first Amateur Championship title.
Also moving up was Cody Shafer, of Evans. He had 66, the day’s low round by three shots. Shafer, who opened with 72-80, is now tied for seventh place, four shots off the lead. He’s tied with former area golfer Chris Waters (74).
Of the other area golfers to make the cut, Austin Vick (72) and Chase Parker (71) are tied with Burton for 13th place at 219, Brian Carter (71) is tied for 22nd place at 220, Jacob Tilton (77) and William Sigg (77) are tied for 56th place at 228.