Gemara

(Mishnah): They would not tie (all four legs of) the Tamid.

31b (Rav Huna or Rav Chisda): This would be a disgrace to Kodshim.

(The other of Rav Huna and Rav Chisda): This is the statute of idolaters.

According to Answer #1, one may tie it with silk or with gold threads. Answer #2 forbids.

Chulin 41a (Mishnah): We do not slaughter above seas, rivers, or Kelim;

One may slaughter into a pit of water, and on a ship one may slaughter onto a Kli (in a way that the blood will run off into the sea).

We never slaughter above an (empty) pit. One may make a pit in his house in order that the blood will flow into it;

One may not do so in Reshus ha'Rabim, lest it encourage Tzedukim in their rituals.

Question: Just like we may not slaughter above seas, lest people think that he serves the angel appointed over the sea, we should forbid Shechitah above a pit with water, lest people think that he serves his reflection!

Answer (Rava): He may slaughter above a pit of water only if it is cloudy. (There is no reflection in it.)

(Mishnah): We never slaughter above a pit...

Question: The Mishnah continues 'one may make a pit in his house... '!

Answer #1 (Abaye): We never slaughter above an empty pit in Reshus ha'Rabim.

Objection (Rava): The Seifa says 'one may not do so in Reshus ha'Rabim.' This shows that the Reisha does not discuss this!

Answer #2 (Rava): Rather, the Mishnah teaches that we never slaughter above an (empty) pit. If one wants to keep his courtyard clean, he makes a place near the pit and slaughters there, and the blood flows into the pit;

One may not do so in Reshus ha'Rabim, lest it encourage the Tzedukim.

Support (Beraisa): If one is on a ship and has no place there to slaughter, he sticks his hand outside the ship and slaughters. The blood runs down the sides of the ship;

We never slaughter above a pit. If one wants to keep his courtyard clean, he makes a makes near the pit, slaughters, and the blood flows into the pit;

One may not do so in Reshus ha'Rabim, due to "do not go in the ways (of idolaters)".

If one did so, we investigate (to see if he is an idolater).

(Beraisa): They used to hit a Korban with sticks (to facilitate the slaughter), like was done in front of idolatry.

Yochanan Kohen Gadol: You offer Neveilos (really, Safek Tereifos) on the Mizbe'ach! (Perhaps the skull was pierced.)

Rishonim

Rambam (Hilchos Shechitah 2:5): We do not slaughter into seas or rivers, lest people say that he serves the water. It looks like he offers to the water. One may not slaughter into a Kli full of water lest people say that he slaughtered to a form that he saw in the water. One may not slaughter into Kelim or a pit, for idolaters do so. If he slaughtered (like this), it is Kosher.

Rambam (6): We may slaughter into cloudy water in which one cannot see a form. One may slaughter outside the pit, and slaughters there, and the blood descends into the pit. One may not do so in the market, lest this strengthen the heretics' rituals. If one slaughtered into a pit in the market, one may not eat from it until we check to see if he is a Min (heretic). One may slaughter over the wall of a ship. The blood runs down the sides of the ship into the sea. One may slaughter over Kelim.

Tosfos (41a DH Ein): R. Tam says that the Isur is to slaughter into a clean pit, for it looks like he receives the blood for idolatry. This is difficult, for if so one need not make a place outside the pit in order to keep his Chatzer clean! Therefore, one should be careful not to slaughter into a pit even if it is not clean.

Bach (YD 12:1): This is not difficult. The Mishnah teaches a solution for a Stam pit, which is clean of earth, but if it is not clean, one may slaughter into it. The Tana did not need to teach this, for it is obvious. This is primary according to the opinion that the Isur to slaughter into a pit is because it looks like he gathers the blood in order to throw it to idolatry. However, the Rashba says that it looks like he gathers it in order to eat over it. The Minim do so. This is even if it is dirty. This is why the Rashba wrote that one may not slaughter into clear water, for if there is earth, it is permitted. He did not say so regarding a pit, for even if there is earth, it is forbidden.

Mordechai (Chulin 607): When one slaughters a Chayah or bird, for which one must cover the blood, he puts loose earth, fills the pit with earth, and slaughters. He may not slaughter into the pit at all.

Beer Sheva (30b DH Lo): Nowadays we are not careful to avoid tying the forelegs and hind legs of an animal together at the time of Shechitah, because idolaters are not particular about this. Or Zaru'a explains that this is why nowadays one may slaughter into a pit even in the market, even though the Gemara forbids this. Also, we can say that the idolaters used to do so only for a Korban. This is the concern!

Poskim

Shulchan Aruch (YD 12:1): We do not slaughter into an (empty) pit, even in the house. If one wants to keep his house clean, he makes an incline outside the pit, and slaughters there, and the blood descends into the pit. One may not do so in the market.

Beis Yosef (DH Ein): The Tur says that it is forbidden in the market, for one is not concerned to keep it clean. The Rashba explains that he makes an incline outside the pit.

Bach (2): The Beraisa says 'if one did so, we must investigate after him.' The Rambam understands that this refers what it says just before this (Shechitah into a pit in the market). This is because even though l'Chatchilah one may not slaughter into a pit in his house or Chatzer, if he transgressed we need not check, for presumably he did so to clean his house or Chatzer. There is no concern to clean the market. Therefore, we must investigate about him. If we do not find that he is Kosher, the Shechitah is forbidden. Why did the Tur write Stam in the name of the Rambam that if one slaughtered into a pit one may not eat from it (until we check)? This connotes that it is even if the pit is in his house or Chatzer. This is not true!

Shach (2): If one makes an incline (outside a pit in his house and slaughters there), it is clear to all that he wants to keep his house clean. One may not do so in the market, for one need not clean the market. Therefore, even if no one sees, it is forbidden, similar to in his house (which discusses when no one sees).

Shulchan Aruch (2): Some say that if one slaughtered into a pit in the market, one may not eat from it until we check to see if he is an Apikoros.

Beis Yosef (DH v'Zeh): The Rashba says that since we must check whether he is a Min, presumably his Shechitah is Pasul due to Safek. Since all the cases were taught together in the Mishnah, all of them are forbidden due to Safek. However, the previous Mishnah says that Shechitah for the sake of mountains is Pasul, and the coming Mishnah says that Shechitah for the sake of a Korban Olah is Pasul. If also the Shechitos in our Mishnah are Pasul, why didn't our Mishnah say so? (It only forbids slaughtering into a sea...) Therefore, perhaps the Shechitah is Kosher (b'Di'eved). We do not establish him to be a Min. He does not lose his Chezkas Kashrus until after we check. This is primary. It seems that Rashi agrees. He says 'we must check him, lest he is a Min, and we will avoid his bread and wine.' I.e. we check to disqualify him, but not to be Machshir him. He is Kosher until we find that he is a Min. It seems that Ba'al ha'Itur holds like the Rambam; the Mordechai holds like the Rashba.

Taz (2): The Maharshal says that if we checked and found that he is a Min, we forbid retroactively even benefit from what he slaughtered into the pit.

Shach (3): One may not eat also from this Shechitah into the pit. The Rashba connotes that the same applies to Shechitah into seas and Kelim. One may not eat from his Shechitah. Even according to the latter opinion that the Mechaber brings, we must check him for the future, also for Shechitah into water and Kelim.

Shulchan Aruch (ibid.): Some permit b'Di'eved, and do not require checking.

Shach (4): This opinion permits this Shechitah without checking. He does not lose his Chezkas Kashrus without checking. We check to disqualify future Shechitos. If we find that he is an Apikoros, retroactively we forbid benefit from everything he slaughtered into a pit, or into water or Kelim.

Rema: Nowadays that idolaters do not do so, we permit b'Di'eved.

Taz (3) and Shach (5): This refers also to Shechitah into seas and rivers and in a Kli (Siman 11). Maharshal brings so in the name of Or Zaru'a.

Shulchan Aruch (28:3): A Koy is a Safek Chayah. One covers its blood without a Berachah. One may not slaughter it on Yom Tov. If he did, he does not cover the blood. If the blood (was absorbed but) is recognized at night, he covers it.

Shach (10): The Tur says that one may not receive the blood in a Kli in order to cover it at night, for we do not slaughter into a Kli. The Drishah says that we do not permit through putting some earth into the Kli, lest onlookers say that the earth is to cover the blood, and they will (think that it is a Vadai Chayah, and) permit its Chelev. The Bach says that nowadays l'Chatchilah one could slaughter into Kelim, like Or Zaru'a says, just we are careful. B'Di'eved, if one slaughtered on Yom Tov, he may receive the blood in a Kli in order to fulfill the Mitzvah of Kisuy ha'Dam. Since letter of the law it is permitted, we need not be careful. It is better to put earth and pebbles in the Kli beforehand. Why didn't the Poskim bring the Bach's Heter? Perhaps he permits only if there is only a little earth, so it does not look like he covers the blood (on Yom Tov). Letter of the law it would be Asur due to concern for idolatry, if not that nowadays idolaters do not do so.

Seridei Esh (2:4 Chimum ha'Behemah, Nispach 3, Sof Michtav 23): They used to hit a Korban with sticks, like was done in front of idolatry. This was discontinued due to Terefah, but not due to Nochri rites! We can say that the Gemara mentioned Terefah, for a verse explicitly forbids it.