Make sure the words you choose after the comma are the thing that was written, not the person who did the writing. The describing phrase should not accidentally describe the author.

Plug each option into the sentence. Does the result have a clear subject right before "received" and a logical meaning? Watch out for options that turn the sentence into a fragment or make the description apply to the wrong word.

The sentence begins with an introductory descriptive phrase: "Written in just six weeks after the author returned from a climbing expedition,".

That phrase describes something that was written. It must logically refer to the word or phrase that comes immediately after the comma. So whatever fills the blank should be the thing that was written, not the person who did the writing.

Ask: What was written in six weeks? It’s clearly the novel, not the author.

So the words in the blank should:

• Begin with the work itself (the novel), not "he" or "the author".
• Function as a noun phrase so it can be the subject of the verb "received".

This means options that start by making the author the subject or that form a full clause (like "he wrote...") instead of a simple noun phrase are suspicious.

Now test each choice in the full sentence for logic and structure:

• Choice A: "Written in just six weeks..., he wrote Summit's Edge—his novel—which received..." This makes the modifier "Written in just six weeks" describe he (the author), which is illogical because the author was not written. It also creates awkward, tangled structure with "which" tacked on.

• Choice C: "Written in just six weeks..., the author's novel Summit's Edge that received..." Here, "that received widespread acclaim" turns "received" into part of a relative clause modifying "Summit's Edge". The whole sentence then lacks a clear main verb after the comma, so it becomes a fragment.

• Choice D: "Written in just six weeks..., the author wrote Summit's Edge and it received..." Now "Written in just six weeks" is describing "the author," again implying the author was written, which is illogical. This is a classic dangling modifier.

All three are ungrammatical or illogical, so they cannot be correct.

The remaining option is B) "Summit's Edge, the author's novel,".

Put it into the sentence:

"Written in just six weeks after the author returned from a climbing expedition, Summit's Edge, the author's novel, received widespread acclaim for its vivid depiction of mountain life."

• The introductory phrase now clearly and logically describes the novel, the thing that was written.
• "Summit's Edge, the author's novel," is a noun phrase that serves as the subject of "received".
• The commas correctly set off the appositive "the author's novel".

This version is logical, complete, and follows Standard English conventions, so B is the correct answer.