I have a link which is union of knots $K_1\cup\ldots\cup K_n.$ I do know how to find link group $\pi_1(\mathbb{R^3}-K_1\cup\ldots\cup K_{n-1})$, for example, using Wirtinger presentation. What I want to do is to find explicit form of the element $[K_n]\in\pi_1(\mathbb{R^3}-K_1\cup\ldots\cup K_{n-1}).$ Is there a way to do that?
I have a feeling that Wirtinger's approach to choosing generators in the link group might not be the best here (at least I don't know how to express loop $K_n$ in terms of Wirtinger's generators).
Thank you.
(EDIT)
I just wanted to add an example in order to try to digest Lee Mosher's answer. Let's consider, for example, Borromean rings $K_1\cup K_2\cup K_3$. Let's label strands of $K_2\cup K_3$ by $a_1,a_2, a_3.$
Then Wirtinger presentation of $\pi_1(\mathbb{R^3}-K_2\cup K_3)$ is group generated by elements $g_1,g_2, g_3$ with single relation $$g_3g_2g_3^{-1}=g_1.$$ Perhaps, we won't need this relation, but still. Generators (or better to say, their representatives) are loops based in some point "above" the drawn diagram, which go each around its personal strand (representative of $g_i$ goes around $a_i$).
What is the elements $[K_1]\in \pi_1(\mathbb{R^3}-K_2\cup K_3)$?
If you are given a planar projection of the whole link $K_1 \cup \cdots \cup K_{n-1} \cup K_n$ then the Wirtinger presentation for $\pi_1(\mathbb R^3 - (K_1 \cup \cdots \cup K_{n-1}))$ is perfect. It has generator for each strand of the planar projection of $K_1 \cup \cdots \cup K_{n-1}$, and the word for $K_n$ is explicitly read off from the sequence of undercrossings of $K_n$ (places where $K_n$ crosses under the strands of $K_1 \cup \cdots \cup K_{n-1}$). Walk yourself through any example of the Wirtinger presentation to see how this works.
Added to address a comment:
I think that what you are missing is the basic statement, and picture, (and proof) of the Wirtinger presentation, in particular the statement, and picture of the generators of that presentation (for the very limited purposes of your post we can ignore relators). You should really consult a proof of the Wirtinger presentation in a very good textbook, such as Rolfsen's "Knots and Links".
Very, very roughly speaking: imagine your eye, looking down, is the base point p for $\pi_1(R^3-(K_2 \cup K_3),p)$. In the picture you provided in your post, there are two maximal subpaths of $K_1$ without an under crossing. Now homotope each of those maximal subpaths (leaving endpoints stationary) by lifting their midpoints allllllll the way up to coincide with p. What you get is a concatenation of two loops based at $p$, namely $g_2$ and $g_1$ (although here I am perhaps careless about the order of the concatenation, and about whether it is really $g_2$ and not $g_2^{-1}$, and similarly for $g_1$).