What is the role of the rope in the story "Rope" by Porter?
"Rope" is a 1928 short story by Katherine Anne Porter, about a marriage that is unfulfilled and bitter.
The nameless man and women -- he and she -- fight over his purchase of a rope instead of the coffee she asked for, and the fight extends throughout the day, with the rope serving as the metaphor for their failure to communicate. Despite this, they reconcile at the end, with the hint that a similar argument will continue.
The rope serves both the literal purpose of coming between them -- she notices it as it has caused the eggs to break, ruining her dinner plans -- and the symbolic purpose of showing their ties of marriage, how they are connected even while angry with each other. The common phrase "hanging by a thread" is useful here, as the rope can symbolize the only thing keeping them together: tied by marriage or they might have separated long ago.
...she reminded him that he had forgot the coffee and had bought a worthless piece of rope. And when she thought of all the things they actually needed to make the place even decently fit to live in ... he couldn't believe it was only a piece of rope that was causing all the racket.
(Porter, Google Books)
Since he has no connection with her personal feelings, he sees the rope itself as the cause of her distress, while she is equally certain that he bought the rope as a direct insult to her and the hard work she does to keep the house in order. Neither of them accept their own fault in the argument, and so this is not likely to be their last fight.