TITLE: Winding number question.
QUESTION [6 upvotes]: In class we defined the winding number as follows: If $\gamma$ is a loop on $\mathbb{R}^2$ that does not pass through a point $p$, the winding number $W( \gamma, p)$ is an integer $n$ that $\gamma$ represents $n$ times the canonical generator in the fundamental group $\pi_1(\mathbb{R}^2\setminus \{p\})$. Essentially, it is thought of as the number of turns $\gamma$ makes about $p$. 
I've been working on problems to better my understanding of this topic. I can't figure out this one, and was wondering if anyone could help me out?
Let $p$ and $q$ be distinct points on the plane and $X = \mathbb{R}^2 \setminus \{p, q\}$. If $\gamma$ is a loop in $X$ such that $W(\gamma, p) = W(\gamma, q) = 0$, does it follow that $\gamma$ represents the trivial element of the fundamental group $\pi_1(X)$?
Thank you so much!

REPLY [5 votes]: Fix a basepoint $x_0\in X$. Let $\gamma_p$ be a loop at $x_0$ going around $p$ exactly once, and $q$ zero times. Similarly, let $\gamma_q$ be a loop at $x_0$ passing around $q$ exactly once and $p$ zero times. Then if for a path $\gamma$, $\gamma^{-1}$ means traverse $\gamma$ backwards, consider the loop $\gamma=\gamma_p\gamma_q\gamma_p^{-1}\gamma_q^{-1}$. Its equivalence class is nontrivial in $\pi_1(X)$ but it has $W(\gamma,p)=W(\gamma,q)=0$.