The Triad Pool

Swap any and every pairs. Never think about how you should divide your portfolio ever again.

Recall that the most well-adopted CPF is two-dimensional where $R_A\times{R_B}=k$. In Sentre, however, the pool can be optionally organized into three tokens. It’s now the pool of triad and the CPF turns to three-dimensional where the third dimension is for $SEN$ as in Fig. 5.

Fig. 5. A visualization of the 3D CPF.

Definition 3. For a pool of $A$, $B$, and compulsory $SEN$, the 3D constant product function is:

(33) $R_A\times{R_B}\times{R_{SEN}}=k$​,

where $k$ is a constant.

To deposit to the pool, a liquidity provider (LP) theoretically needs to divide their portfolio into three equal portions regarding value. However, LPs never worry about this due to Simulated Single Exposure which allows people to deposit even on one side (see Asymmetric Deposit). Especially, although the formula has a higher dimension, the formula isn’t much different from the 2D CPF in trading.

When a trader swaps $A$ from $B$, for example, the third token, which is $SEN$​ in this case, will be ignored. The formula is boiled down to $R_A\times{R_B}=\frac{k}{R_{SEN}}=k'$​, where $k'$ is a constant as well.

Assume a trader swaps $r_A$ for $r_B$, the newest state of token $A$ would be $R'_A=R_A+r_A$​. Because of the CPF, we have:

(34) $R'_B=\frac{R_AR_B}{R'_A}=\frac{R_AR_B}{R_A+r_A}$.