Tutte polynomial of a cycle

Determine the Tutte polynomial for the cycle of length $k$.

Hint
Use the recurrence for the Tutte polynomial $ T_G (x, y) $.
\[T_G = \begin{cases} x\cdot T_{G-e} & \text{when $e$ is a bridge,} \\ y\cdot T_{G-e} & \text{when $e$ is a loop, } \\ T_{G-e} + T_{G/e} &\text{when $e$ is not a bridge nor a loop.} \end{cases} \]
Furthermore, an edgeless graph has $ T_G = 1 $.
Solution
In the first step, the last option is used. So $T_{C_k} = T_{P_{k-1}} + T_{C_{k-1}}$.

By removing leaves, we find that $T_{P_{k-1}} = x^{k-1}$, which leads to the recurrence $T_{C_k} = x^{k-1} + T_{C_{k-1}}$

Lastly, we apply recurrence to the bigon, i.e. to two vertices connected by two edges. Then one vertex is left with a loop $ C_1 $, where $ T_ {C_1} = y $.
Answer
The Tutte polynomial of a cycle is: $ T_{C_k} (x, y) = x^{k-1} + x^{k-2} + \cdots + x^2 + x + y $.