It's called Knowledge and it's a short story by John Frizell in Nature. Did you know that Nature publishes short stories, one at the back of every issue? They do!
No human could have grasped the squid's name. Human eyes could not distinguish the differences in shades of colour or register the intervals at which they changed to define the unique pattern that was his name. The squid was concentrating hard because he was holding two conversations at once, one deliberately misleading, the other closer to the truth, as he glided through the deep ocean, his mantle pulsing gently, powering him with puffs of water.

On his right side the flowing patterns of colour were attempting to persuade a female to mate with him. He was being as persuasive as possible without exaggerating; females were very good at picking up on male lies, and like all female giant squid she was considerably bigger than him, or any adult male.

On his left he was chatting to another male about recent sightings of Loud Shadows, the huge predators that were the only things in the ocean that attacked giant squid.
By short story, I mean extraordinarily short. You can finish it in a handful of minutes. Do go read it; it's a clever idea well-told! 

Really, read it now, because below this picture of a giant squid I am going to spoil it to pieces.

I love the way science fiction can start from realistic, plausible concepts and transition seamlessly into jumping off the deep end. Here's a breakdown of the science in "Knowledge," from plausible to deep end.

The story's first scientific concept is that giant squid communicate with skin patterns, and furthermore, that they can communicate two different ideas at once with the skin on each side of their body. This "split communication" is a hallmark of cuttlefish, but to my knowledge has never been reported from squid--yet.

The second major concept seems rather more far-fetched: "hatched memories." The idea of inheriting memory from past generations has intrigued scientists and writers alike for years (possibly generations!) and there is in fact some curious and controversial research on the subject. The level on which we truly inherit memory (if indeed we do at all) is likely to be far more subtle than the "hatched memories" of the giant squid in Knowledge.

The third concept, the way sperm whales (Loud Shadows) hunt giant squid, returns very close to reality. The author suggests that sperm whales use one kind of sonar for locating the squid and another (the Killing Sound) for stunning it. As I've written before, this is entirely plausible but not yet proven.

The story's closing concept--our hero squid invents writing--is a little Clan of the Cave Bear in terms of a single individual inventing something that would probably develop gradually with contributions from many individuals. But the lone inventor makes for better fiction, and in any case, the squid's creation is based on a very real aspect of the deep sea: its pollution with Transparent Death, or old fishing lines.

What will happen once giant squid can record and share information? A cultural renaissance? A powerful new civilization? It would be fun to give this story to students, then have them write a sequel or epilogue.