Knot I, Excelsior. Two knights discuss the distance they will have travelled that day, uphill and downhill at different speeds. The older knight obscurely explains the mathematical problem.
Carroll's Solution: As with most of the Knots, the solution includes: a simplified restatement of the problem, a method to arrive at the solution, the solution, a discussion of readers' solutions, then readers' grades. In his discussion, Carroll relates that one reader accuses the senior knight of untruthfulness (this is rebutted by Carroll, using the knight's tone). Another reader answers the problem by extending the story (this is quoted). The poem of two readers answering the problem is also quoted.
Knot II, Eligible Apartments. Professor Balbus, named after a hero with "anecdotes whose vagueness in detail was more than compensated by their sensational brilliance", is given a problem by students. The number of guests for a party is described in puzzling terms. He in turn creates a mathematical problem for them: two answers are required of readers.
Solution: The mathematical problem is solved with the aid of a diagram. Those employing "guesswork" are given partial credit. One reader suggests the genealogical problem can be solved by "intermarriages", to which Carroll replies, "Wind of the western sea, you have had a very narrow escape! Be thankful to appear in the Class-list at all!"
Knot III, Mad Mathesis. Overbearing aunt Mad Mathesis bets her niece that she can select a train from London that will pass more trains than her niece's does. The niece loses, but thinks she has found a solution to win, a second time.
Knot IV, The Dead Reckoning. The two knights of Knot I, in a modern guise, are party to a dispute about the weight of passengers' bags lost overboard from a ship.
Knot V, Oughts and Crosses. The aunt and niece from Knot III are in an art museum. Trading snipes as before, the aunt evades her niece's logical problem: The niece's preceptress had told her girls, "The more noise you make the less jam you will have, and vice versa." The niece wants to know if this means that if they are silent, they will have infinite jam. Instead, her aunt responds with her own logical problem.
Knot VI, Her Radiancy. Two travellers appear in Kgovjni, a land referenced in earlier Knots. The ruler places them in "the best dungeon, and abundantly fed on the best bread and water" until they resolve a logical problem.
Solution: Two problems are posed, the first of which is resolved by word-play. Much later, after the solutions to Knot VII, Carroll returns to Knot VI, to describe the second problem in detail, and to rebut criticisms of readers (who were identified by name) that they were duped.
Knot VII, Petty Cash. The aunt and niece encounter "by a remarkable coincidence" others who are travelling not only on the same train, but at the same station, on the same day, at the same hour. Lunch bills are muddled due to the aunt's reluctance in writing down numbers that could "easily" be memorised.
Solution: Carroll gives a solution which "universally" produces an answer, then gives detailed critiques of several other approaches that only "accidentally" give a solution.
Knot VIII, De Omnibus Rebus. The travellers of Knot VI are leaving Kgovjni with relief, when a mathematical problem occurs to one of them.
Knot IX, A Serpent with Corners. The characters of Knot II, and Balbus and his two students, return to give three problems loosely connected by a narrative.
Knot X, Chelsea Buns. Mad Mathesis and her niece return, as well as Balbus and his two students.
Solution note: The puzzle called The Change of Day is never answered, as Carroll is "waiting for statistics" and is himself "so entirely puzzled by it."

Question: Who is in Knot IX with their two students?
Answer: Balbus