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In the yom kippur musaf my community recited a poem going through the whole story of the deaths of the "asara harugai malchus" the ten martyrs. It is a "חטאנו" poem recited before the confession section.
Why on yom kippur do we mourn their deaths, seemingly it has no specific connection to yom kippur? We should mourn their deaths on tisha b'av?
Rashi Bemidbar 20:1
Why was the death of Miriam juxtaposed to the parah adumah? To teach us that just like sacrifices atone, so too does the death of the righteous.
I assume (and, if I remember correctly, Rav Soloveitchik says this in his machzor) that the deaths are being invoked as some kind of kapparah.
Although I don't have the time right now to delve into this long article right now, it seem to spend over 20 pages addressing your question. One line that sticks out to me is:
Arzei ha-Levanon by clear allusion identifies the Martyrs as Yom Kippur sacrifices: ",לב טהור יש חי קדשי םקדשי טתן חמורה במיתה" “Pure of heart, holiest of holy ones, Kodshei Kodashim, their slaughtering in a severe death,” paraphrasing Mishnah Zevahim 5:1: איזהו של מקומן שחיטתן קדשי םקדשי זבחי םשחיטתן הכפורים יום של ושעיר פר בצפון בצפון, “Kodshei Kodashim, their slaughtering is in the north, the ox and goat of Yom Kippur, their slaughtering is in the north.
