Homer's "Odyssey"? Homer's "The Odyssey"?

One can readily find these stats at Google Boooks (no vanilla Google).

The moral:

When getting short

"on Homer's Odyssey" About 47,800 results

"on Homer's The Odyssey" About 1,470 results

be really short (no "the")

_

Now, when getting longer

"on Odyssey by Homer" 4 results

"on The Odyssey by Homer" About 3,270 results

be really long, and use the "the." :-)

The editions of Homer (translated by Richmond Lattimore) that I read in college were rendered as The Iliad and The Odyssey. Supposing that those are the names you want to use for these two works in your paper, you may be interested in this style recommendation from the Chicago Manual of Style, fifteenth edition (2003):

8.179 Full and shortened titles. A title cited in full in the notes or bibliography may be shortened in the text. A subtitle may be omitted, or an initial a, an, or the may be dropped if it does not fit the surrounding syntax. ...

[Relevant example (1):] Hawking, in A Brief History of Time, opens up the universe.

[Relevant example (2):] Hawking's Brief History of Time explains black holes with alarming lucidity.

This style advice endorses identifying The Odyssey with or without the initial The depending on how your sentence reads. For example,

Homer, in The Odyssey, juggles themes of determination, struggle, loyalty, and revenge.

or

Homer's Odyssey juggles themes of determination, struggle, loyalty, and revenge.

You don't have to accept Chicago's advice on this point, of course, but it's available as an option if you choose to follow it.