The answer to your question would be that the definition that best identifies and explains the function of the word "tying" as it is used in this sentence is the following one: In the selected sentence, the word "tying" is a verbal that functions as an adverb.
A verbal is a word formed from a verb but functioning as a different part of speech. "Tying his shoe" is a participle phrase. In fact, it is present participle phrase, as it is created from the form of a verb used with the verb to be as an auxiliary verb (progressive tense). Removing the auxiliary verb and using the -ing form of the main verb produces a present participle. In this case the participle is functioning as an adverb because it is telling something about the manner in which Garrett prevented an embarrassing fall. He did this tying his shoe. You can tell this by asking the following question: How did Garrett prevent the embarrassing fall? Tying his shoe.
