First homology group of the complement of a twisted circle in 3-space

Question

Let $\mathbb{S}^1_0$ be a circle in $\mathbb{R}^3$ with $n$ number of full twists (rotation by 360). Let $N(\mathbb{S}^1_0)$ be a tubular neighborhood of $\mathbb{S}^1_0$. The first homology group $H_1(\mathbb{R}^3 \setminus N(\mathbb{S}^1_0))=\mathbb{Z}$. Assume that the loop in $\mathbb{R}^3 \setminus N(\mathbb{S}^1_0)$ that is identified to $\mathbb{S}^1_0$ is $[1]$ or $[0]$ in $H_1(\mathbb{R}^3 \setminus N(\mathbb{S}^1_0))$.

Let $\mathbb{S}^1_1$ be a circle in $\mathbb{R}^3 \setminus N(\mathbb{S}^1_0)$ with $n+1$ number of full twists. How to describe this circle $\mathbb{S}^1_1$ in $H_1(\mathbb{R}^3 \setminus N(\mathbb{S}^1_0))$?

Answer 1

It is just the definition of the linking number $\text{lk}(K_0,K_1)=n+1$.