Karthspire is a cave in The Elder Scrolls V: Skyrim. It is located east of Markarth in The Reach and within a mountain nestled in the river valley that contains Karthspire Camp. Upon arrival to Karthspire, the Dragonborn will be greeted by Forsworn that surround the entrance.
A Forsworn Briarheart and several weaker Forsworn have set up camp in the first room.
The first puzzle is the one to lower the bridge to get to the other side of the room. If Delphine or Esbern are following the Dragonborn, they will reference the cave as being "definitely Akaviri stonework". To solve the puzzle, turn the symbols to the Dragonborn symbol, the one shaped a little bit like a heart.
The second puzzle is with labeled pressure plates. Flames will shoot up through the holes in the panels if the Dragonborn steps on any pressure plate, other than the plate with the Dragonborn symbol. Activating the chain at the pillar in the back of the room will disable the floor traps. Any followers, other than Delphine or Esbern, should wait. One may also cross this area using either Whirlwind Sprint or Become Ethereal. The perk Light Foot works here, so a stealth-oriented Dragonborn shouldn't have much trouble.
Solving the third puzzle will allow entrance to the Sky Haven Temple. To complete this puzzle, the Dragonborn must be accompanied by Esbern. Upon activating the circle on the floor, the Dragonborn cuts their hand over a giant Blood Seal symbol and the door to Sky Haven Temple will then open.