Gemara

(R. Elazar): "Alei Zayis Toraf b'Fiha" - the dove 'said' 'I prefer bitter food (like olives) directly from Hash-m, to food sweet as honey given by man.'

Avodah Zarah 39a: A piece of Chiltis (a spice, of Nochrim) is forbidden because they cut it with knives that absorbed forbidden taste.

Even though (a Stam Keli is not a Ben Yomo, i.e. it was not used for Isur in the last 24 hours), and we permit Nosen Ta'am li'Fgam (absorbed tastes over a day old, for they are detrimental), because Chiltis is (so) sharp, it causes the absorbed Isur (in the knife) to become beneficial.

Chulin 111b (Rav Kahana): One may not put hot food in a basin in which meat was salted. A radish cut with a knife (used for meat) may be eaten with milk.

(Abaye): The radish is permitted because it absorbed permitted Ta'am. The basin absorbed forbidden Ta'am (blood).

Objection (Rava): Also the taste absorbed in the radish becomes forbidden when it mixes with milk! Rather, the radish can be permitted if one tastes it (and does not taste meat in it). One may not taste food put in the basin.

36b (Rabanan): Rava permits Indian ginger jam. (He is not concerned for forbidden absorptions.)

Rishonim

Rosh (Teshuvas 24:7): My Rebbeyim permitted an onion cut on Pesach with a knife used to scrape dough from the kneading bowl. If an onion was cut with an Asur knife, Sefer ha'Terumos requires peeling off a layer, for the sharpness makes the absorptions beneficial, like we say about Chiltis. A knife used only for cold is different, for cold usage does not absorb or emit.

Poskim

Shulchan Aruch (447:8): If one cut olives with a new knife, even if he pickled them in a used bowl, if it is not Ben Yomo, all permit them.

Tur: Sefer ha'Terumos forbids a radish or onion cut with a Nochri's knife, even if the knife was not Ben Yomo, for he equates this to a stick of Chiltis, and we say that the sharpness sweetens the fatness, and makes its taste beneficial. Likewise, if olives were pickled during the year in a used (Chametz) Keli, the olives' sharpness makes the Keli like a Ben Yomo. Bitul does not apply, for we estimate based on the entire knife (perhaps it is saturated with Chametz). If so, the olives are like Chametz. The Rosh said that even Sefer ha'Terumos would permit if the knife was used only for cold, but not a normal knife. Rabeinu Meir mi'Rotenberg ruled like Sefer ha'Terumos, but he leaned to be lenient and did not protest against those who were lenient. I say that even Sefer ha'Terumos would agree that if one cut olives with a new knife, even if he pickled them in a used bowl, if it is not Ben Yomo, it is permitted. Since we add water at the time of pickling, their sharpness is blunted and the absorbed taste does not become beneficial.

Question (Taz 14): Why don't we estimate if there is 60 times the volume of the knife? Surely, all the olives together are 60 times its volume. Here, at the time they absorbed, Chametz was permitted. This is unlike a Nochri's knife. The Tur's final words connote that if he cut with used knife, there is no Heter through Bitul of what was absorbed in the knife. We must say that the olives were cut before pickling, for then they are sharp. The Tur said that the water blunts the sharpness. Isur v'Heter ha'Aruch says that we are stringent only if they were cut after pickling. The Tur says oppositely! It seems that Isur v'Heter cited Orchos Chayim, and not (Tur) Orach Chayim. Bitul in 60 does not apply, for each olive absorbed Chametz before pickling, and it did not have 60 times as much Heter as the knife. When they are pickled, nothing brings taste from one to another to join them together. When onions are cooked together, the gravy brings taste from one to another. When unripe grapes are pounded together, they mix and join to be Mevatel taste emitted from the mortar. How did the Tur bring a proof from an onion cut on Pesach? This is unlike pickling before Pesach (for Chametz can be Batel before Pesach!) It seems that he proves only that the acidity of olives equates them to Chiltis. The Rosh normally permits Nosen Ta'am li'Fgam. He forbids onions because they are sharp (and the same applies to olives).

Magen Avraham (39): Even though sharp foods forbid even if it is not Ben Yomo, here the water is Mevatel the sharpness. The Shach (YD Sof Siman 70 and 114:13) requires a majority of water. The Rema connotes otherwise. I say that it depends on the Rav's estimation. If the bowl is a Ben Yomo it is forbidden, for pickling is like cooking.

Gra (32): We permit (Avodah Zarah 39b) pickled olives of Nochrim. We are not concerned for absorptions of Isur.

Mishnah Berurah (89): We forbid if they were cut with a used knife, even if if it was clean and not a Ben Yomo, if they were cut before pickling or after they finished pickling, for then they are sharp. They are not sharp from when water is put in them until they finish pickling.

Kaf ha'Chayim (200): This answers the question of the Taz and Magen Avraham (below).

Mishnah Berurah (89): During Pesach one may drink the water in which they were pickled, even if they were cut with a Ben Yomo knife, as long as they are pickled before Pesach, for the Chametz is Batel in the water. If they were cooked before Pesach, even the olives are permitted, for they emitted the Chametz. Some forbid. One may be lenient if needed for Simchas Yom Tov if he knows that the knife was clean.

Kaf ha'Chayim (201): Chak Yosef is stringent about the water, for a Stam knife is not clean and the olives absorbed actual Isur (not just a taste), and they become like Nevelah and forbid the water.

Mishnah Berurah (90): During pickling they are not sharp, i.e. if there is more water than olives. The Shulchan Aruch connotes that if they were left pickling in a Ben Yomo bowl more than 24 hours, they are forbidden, for this is like cooking. Even though the absorbed taste is Pagum after the 24 hours, since the olives are a little sharp, one should be stringent.

Kaf ha'Chayim (206): The Magen Avraham forbids only if they were put in the bowl right after the Chametz was removed, so they finish pickling at the same time that the taste becomes PAgum. The Shulchan Aruch connotes differently, and so say most Acharonim. Beis Hillel say that a Keli is not a Ben Yomo only if it was empty for 24 hours.

Mishnah Berurah (91): We are stringent about Nosen Ta'am li'Fgam during Pesach, so we permit only if they were removed from the Chametz bowl before Pesach.

Kaf ha'Chayim (198): We are lenient also if they were cut with a clean knife that was used only with cold Chametz.

Kaf ha'Chayim (199): In YD 96:1, the Mechaber favors the opinion that sharpness does not forbid if the knife was not a Ben Yomo. We are more stringent about Pesach, for some forbid Nosen Ta'am li'Fgam during Pesach, even though the Mechaber does not rule like them.

Shulchan Aruch (YD 96:4): Lemon juice of a Nochri is permitted.

Shach (20): Shibolei ha'Leket says that it should be forbidden, for he cuts with his knife, and the lemon absorbs due to its sharpness. We must say that the taste in the knife is Batel after the first lemon or two (and they are Batel in the juice of the many lemons afterwards). The Bach says that he must hold that sharpness sweetens the absorbed taste only for Chiltis. Therefore, we are concerned only for fat on the knife, but not for absorptions in the knife (they are detrimental). We hold like Sefer ha'Terumos (that all sharp foods sweeten Pagum tastes), like the Tur says about olives, so also the lemon juice is forbidden.

Shulchan Aruch (114:8): Pickled olives of Nochrim are permitted if they were not cut with their knife.

Rema: Since they are sharp, they absorb from the knife. If they were pickled in Nochri Kelim, they are permitted, for the water blunts their sharpness.

Shach (12): Isur v'Heter says that a Nochri knife forbids them only if they were cut after pickling, for then they are sharp.

Rebuttal (Magen Avraham OC 447:38): In OC 447, we forbid if they were cut with a Nochri's knife even before pickling. The Tur says that water is Mevatel the sharpness! Rather, we discuss before pickling. Each olive absorbs a little Chametz from the knife, and does not emit it. Foods emit in cooking, but not in pickling. This is why we Kasher through boiling, but not through pickling. Olas Shabbos says that this is like the opinion that Chozer v'Ni'ur (after Bitul, an Isur can become forbidden again), but the Halachah is unlike this. This is wrong. The Tur does not say Chozer v'Ni'ur, yet he forbids the olives! We must say like the Shach (96:20). I say that this is like meat that fell into many pots, one after another (Rema 98:4. We need 60 times its volume in each pot to permit the pot.) They do not join (we do not say that it emitted part of its taste in the first pot.) All permit the water in which the olives were pickled, for the Chametz is Batel in 60. Normally, a piece that became Asur becomes (totally Asur) like a Neveilah, and we need 60 times all the pieces. We do not say so here, for before Pesach it was Heter. If the olives were cooked before Pesach, they become permitted, for the Chametz is emitted, and it is Batel in 60. The Shulchan Aruch (YD 106:1) forbids, but Maharam Lublin (28) says that this is only when it absorbed Isur. Here, it absorbed Heter, so even the piece is permitted. One may keep them on Pesach, but he may not eat them, for the Isur is not totally emitted. We are lenient about meat, for a tiny amount does not forbid, but any amount of Chametz is forbidden.

Rema (122:3): Some say that Nosen Ta'am li'Fgam does not apply to a sharp food. Therefore, if one cooked a sharp food in a forbidden pot that was not a Ben Yomo, e.g. a dish that is mostly vinegar or spices or other sharp things, it is forbidden. It is not considered sharp due to a small amount of spices.

Gra (9): The Gemara forbids Chiltis and radish even if the knife was not a Ben Yomo.

Gra (10): Why does Rava permits Indian ginger jam? Ginger is sharp! Semag says that a sharp dry food does not sweeten Pagum tastes. However, Chiltis is dry, yet it sweetens Pagum tastes! The Mordechai says that Nochrim use a special knife to detach it, or they detach it by hand. They cook it, but Stam Kelim of a Nochri are not Ben Yomo. They cook it with honey, and the ginger is the minority, so it is not considered sharp.