Kirk is attacked and clearly stabbed in the lower back. McCoy later reports that the knife narrowly missed Kirk's heart and Kirk is bandaged across the upper chest, nowhere near where the knife wound occurred.
When attacking the ship, Kirk orders torpedoes 2, 4, and 6 to be fired, however only two torpedo shots are shown.
When Sarek and Amanda greet the Enterprise officers for the first time, Spock is wearing his dress uniform shirt. In the next scene, Kirk is still in his dress shirt, but Spock is wearing his regular uniform shirt. The next time we see Spock, he is wearing the dress uniform again.
During Kirk's and Thelev's fight, Thelev stabs Kirk on the RIGHT side and Kirk places his right hand over his right side. After the fight ends, Kirk calls Spock on the wall intercom, and Kirk is covering his left side with his left hand and there is blood on his left hand. Dr. McCoy later reports the stab wound is to the LEFT lung, but Kirk was stabbed on the RIGHT side well below the rib cage.
When the 'Andorian' is on the bridge, all close-up shots of him and Kirk show the two guards behind with their hands at their sides. Then the shots from further away show their hands behind their backs. This alternates for quite a few scenes.
The distances and speeds associated with the attacking ship make no sense. When it first appears, Chekov says that is out of phaser ranger and just at the limit of sensor range. This would place the ship fairly close to the Enterprise, yet when it begins to approach, Chekov says the ship is traveling at "almost warp 10." Warp 10 is faster than any craft can travel, so even a split second at this speed would place it hundreds of light years past the Enterprise. Later, when the ship is attacking again, it's going warp 8, yet once again, it takes a long time for it actually to reach the Enterprise.
The enemy ship is repeatedly shown moving when it's supposedly moving at almost warp 10 and then at warp 8, but any warp speed is faster than the speed of light, meaning that the ship would be completely invisible to the human eye, just as the movement of light is invisible to the human eye.
Update: Warp 10 meaning infinite speed is from TNG. In TOS, this was not the case and in several cases, ships traveled faster than that.
The enemy ship is repeatedly shown moving when it's supposedly moving at almost warp 10 and then at warp 8, but any warp speed is faster than the speed of light, meaning that the ship would be completely invisible to the human eye, just as the movement of light is invisible to the human eye.
Update: Warp 10 meaning infinite speed is from TNG. In TOS, this was not the case and in several cases, ships traveled faster than that.
Doctor McCoy diagnoses Sarek with a bad heart valve. The Vulcan heart is located on the right side, mid-torso. During the surgery on Sarek, Doctor McCoy is operating on Sarek's left side, near his waist.
Spock and Sarek were alert and talking after the transfusion and surgery way faster than they should have been. Dr. McCoy could not have finished the surgery too much before Kirk had finished off the alien ship that was attacking. Kirk went to sickbay as soon as he was finished on the bridge so Spock and Sarek really had no time for recovery.
When the Vulcans arrive, we have a clear view of Shuttlecraft NCC-1701/7 "Galileo". Although this shuttle was destroyed in The Galileo Seven (1967), "Enterprise" would have put in for re-supply in the time since then, thus the "Galileo" was simply replaced.
When Kirk collapses after his attack, the wall behind him quivers.
As Sarek and Amanda leave the reception, tape can be seen on the floor designating the actors' marks for the scene.
Doors are supposed to open automatically when someone approaches, yet Spock stands at the door in his quarters and touches the door itself, yet it doesn't open. His mother has just left, so we know the door works. How do the doors know not to open this time?
After Thelev stabs Kirk in the back, Kirk flips Thelev over his shoulder, and in the process Kirk's back is seen, with no sign of a wound anywhere.
During the fight in the corridor, one of Kirk's moves was to launch himself at the Andorian and kick him with both feet. However, when he does this, his feet actually hit the wall, missing the Andorian.
McCoy states that Sarek's heart rate is over 300. However, the heart monitor sound effect is that of a normal human heart rate.
When Chekov first calls Kirk from the bridge in the opening sequence to inform Kirk that the shuttlecraft is on final approach, we hear the familiar bosun's two-note whistle, and Kirk answers immediately at the nearest intercom without knowing that the call is for intended for him. There should have been the whistle, followed by Chekov's voice, filtered, calling 'Bridge to Captain Kirk. Bridge to Captain Kirk' making it clear that Chekov wanted to talk to Kirk rather than someone else.
Spock's argument - that the safety of the ship and the various delegations aboard her are his primary responsibility - could just as easily be turned against him. Sarek is not merely Spock's father; he is a Federation ambassador, and an important one. One could argue that Spock's unique ability to help the ambassador would supersede his general responsibility to the ship and passengers. Furthermore, he could have put Scotty in command of the ship while he was in surgery, even though Scotty does not appear in this episode, it could have been referred to. This had already been done in This Side of Paradise (1967)(#1.24).
The Orion ship was on a suicide mission. The Orion ship could travel at warp ten. The Orion ship could simply fly through the Enterprise at warp ten to destroy the Enterprise and all aboard. There would be no need to have an agent onboard the Enterprise or for the Orion ship to waste time firing phasers at the Enterprise.
Kirk was previously unaware that Spock is Sarek's son. This information would certainly be in Spock's personal record, and while Kirk would not be expected to be familiar with the details of every crew member, he must surely have looked at the files of his bridge crew and in particular his First Officer.
When security brings Thelev, the Orion posing as an Andoran, to the bridge, his hands are not bound and he is allowed to stand right next to the already injured Captain Kirk, with the guards standing some distance away near the door and not holding weapons. What was to prevent Thelev from physically attacking Kirk? He had already demonstrated his vicious fighting ability.
Kirk orders Security to the brig. If there is one place Security would already be, it's the brig, especially with a prisoner.
When Sarek arrives on board, he exits the hangar deck, but the black sign to the left is misspelled "Hanger Deck."
After the Orion ship is destroyed, and Thelev's suicide, Kirk leaves the bridge to head for the sick bay. Before he does, he tells Mr. Chekov to take over. Why would Kirk leave Chekov in command when Chekov is only an ensign? The helmsman, navigator, and Uhura are all Lieutenants, so one of them should have been left command of the bridge.
When Dr. McCoy is chatting with Amanda at the reception, he refers to her husband as "Surak" rather than Sarek.
Sarek's blood pressure is noted as being 90 over 40. Dr. McCoy comments that he wish he knew if that was good or bad. Previously, after giving Spock a physical he says that his blood pressure is almost nonexistent, in other words, perfectly normal for a Vulcan.