Linking number and Homotopy This statement has been taken from Wikipedia page. " homotopy classes of a curve in 3-space minus a circle are determined by linking number.
It is also true that regular homotopy classes are determined by linking number, which requires additional geometric argument."
Intuitively I feel it's correct, I am looking for rigorous proof of it. Can someone suggest some reference for it?
 A: This is Alexander duality.  $H_1(S^3-S^1)$ is isomorphic to $H^{3-1-1}(S^1)\cong\mathbb{Z}$.  If you work through how the isomorphism works, you can get this: every homology class $[\alpha]\in H_1(S^3-S^1)$ can be represented by an oriented closed (multi)curve $\alpha$ in $S^3$.  Since $H_1(S^3)=0$, then $\alpha$ is the boundary of a 2-chain.  In a 3-manifold, we can assume this 2-chain is represented by an embedded oriented surface $\Sigma$ with $\partial\Sigma=\alpha$ if we want (see Hatcher's book on 3-manifolds), and furthermore by the usual density argument we may assume $\Sigma$ is transverse to the $S^1$, so it intersects $S^1$ at finitely many isolated points.  Let's define a $1$-cochain $\phi$ for $S^1$ by, for $\beta$ a $1$-chain for $S^1$, setting $\phi(\beta)=\Sigma\cdot\beta$, where $\Sigma\cdot\beta$ is the algebraic intersection number between $\Sigma$ and $\beta$.  This defines a cohomology class in $H^1(S^1)$.  In particular, the isomorphism $H^1(S^1)\cong\mathbb{Z}$ can be given by $[\phi]\mapsto \phi([S^1])$ where $[S^1]\in H_1(S^1)$ is the fundamental class.  Thus, $H_1(S^3-S^1)\cong\mathbb{Z}$ by sending $[\alpha]$ to $\Sigma\cdot S^1$.  If you recall, this is one of the definitions of the linking number between two curves.
If you're instead familiar with the diagrammatic way to count linking numbers, then the correspondence comes from constructing $\Sigma$ via Seifert's algorithm.  The algebraic intersection number calculation can be localized to the crossings between components, which is the only place where $\Sigma$ might intersect the other component.  (See Rolfsen's "Knots and Links" section 5.D for eight equivalent formulations of the linking number.)
Dually, we could have chosen such an oriented surface $\Sigma$ with $\partial\Sigma=S^1$, and then $H_1(S^3-S^1)\cong\mathbb{Z}$ by $[\alpha]\mapsto\Sigma\cdot\alpha$.  This $\Sigma$ is known as a Seifert surface for the knot, and its function is to calculate linking numbers with the knot through the algebraic intersection number with the surface.  You can think of a Seifert surface as a kind of branch cut of a "logarithm" that counts the number of times you've gone around the knot.
Yet another way of thinking about this is that $H_1(S^3-S^1)$, by the universal coefficient theorem, is isomorphic to $H^1(S^3-S^1)$, and first cohomology with  $\mathbb{Z}$ coefficients is represented by the functor $[-,S^1]$ taking a space to homotopy classes of maps to $S^1$.  Given a $\phi:S^3-S^1\to S^1$ that generates the cohomology group, the isomorphism $H_1(S^3-S^1)\cong\mathbb{Z}$ is given by taking a loop $\alpha:[0,1]\to S^3-S^1$ and looking at the composite $\phi\circ\alpha:[0,1]\to S^1$ and calculating its winding number.  You can choose a smooth representative $\phi:S^3-S^1\to S^1$ that generates the group and then consider the level set of a regular value, which is a closed oriented surface $\Sigma$.  The winding number can equivalently be calculated via the algebraic intersection number between $\Sigma$ and $\alpha$.
