Let ¥textit{q} be a positive real number. The left and right ¥textit{q}-deformed rational numbers were introduced by Bapat, Becker and Licata via regular continued fractions, and the right ¥textit{q}-deformed rational number is exactly ¥textit{q}-deformed rational number considered by Morier-Genoud and Ovsienko¥cite{MO1}, when ¥textit{q} is a formal parameter. They gave a homological interpretation for left and right ¥textit{q}-deformed rational numbers ¥cite[Theorems 4.7 and 4.8]{BBL}. In this talk, we give a formula for computing the ¥textit{q}-deformed Farey sum of the left ¥textit{q}-deformed rational numbers based on the negative continued fractions. We use this formula to give a combinatorial proof of the relationship between the left ¥textit{q}-deformed rational number and the Jones polynomial of the corresponding rational knot which was proved by Bapat, Becker and Licata using a homological technique. We combine the homological interpretation of the left and right ¥textit{q}-deformed rational numbers and the ¥textit{q}-deformed Farey sum, and give a homological interpretation of the ¥textit{q}-deformed Farey sum. [BBL] Bapat, A., Becker, L., Licata, A. M.: $q$-deformed rational numbers and the $2$-Calabi--Yau category of type $A_2$, Forum Math. Sigma 11 (2023), Paper No. e47, 41 pp. [MO1] Morier-Genoud, S., Ovsienko, V.: $q$-deformed rationals and $q$-continued fractions, Forum Math. Sigma 8 (2020), Paper No. e13, 55 pp. [XR2] Ren, X.: On $q$-deformed Farey sum and a homological interpretation of $q$-deformed real quadratic irrational numbers,arXiv:2210.06056, 2022.