Kingdom through Covenant by Wellum and Gentry, 

“Animals are slaughtered and sacrificed. Each animal is cut in two, and the halves are laid facing or opposite each other. Then the parties of the treaty walk between the halves of the dead animals…What is being expressed is this: each party is saying, “If I fail to keep my obligation or my promise, may I be cut in two like this dead animal.” The oath or promise, then, involves bringing a curse upon oneself for violating the treaty. This is why the expression “to cut a covenant” is the conventional language for the initiation of a covenant in the Old Testament

Abrahams covenant: If I fail to keep my obligation, may I be cut in two like this dead animal, then involves bringing a curse upon oneself for violation. Sign: circumcision. 

The covenant: Jesus sacrificed himself, For the new covenant. God will not turn from His Covenant!  He cannot cut himself off. So continually plead for Mercy because I am in covenant. Admit complete dependency on Grace (not myself) Only Grace can save-it is not in me. Sign: Baptism and Blood of Christ.