Your intuition is correct: the map p : L &rarr; <i>S</i><sup>1</sup> is not nullhomotopic.<br>

If p : L &rarr; <i>S</i><sup>1</sup> was nullhomotopic, then it would factor through the universal cover &#8477; of <i>S</i><sup>1</sup>.<br>
The inclusion &iota; : L &rarr; <i>S</i><sup>1</sup> &times; <i>S</i><sup>1</sup> would therefore factor through &#8477; &times; <i>S</i><sup>1</sup>.<br>
Let &iota;' : L &rarr; &#8477; &times; <i>S</i><sup>1</sup> denote a lift of &iota;,
and let L<sub>0</sub> := &iota;'(L).<br>
The subset L<sub>0</sub> &sub; &#8477; &times; <i>S</i><sup>1</sup> is then one of the connected components of the preimage of L in &#8477; &times; <i>S</i><sup>1</sup>.<br>
Let us call the other components L<sub><i>n</i></sub> for <i>n</i> &isin; &#8484;.<br>

Let <i>C</i><sub>0</sub> &sub; &#8477; &times; <i>S</i><sup>1</sup> be a loop in that
separates L<sub>0</sub> from L<sub>1</sub>.<br>
We may also assume that <i>C</i><sub>0</sub> is not nullhomotopic in &#8477; &times; <i>S</i><sup>1</sup> [this still needs a small argument...].

The projection <i>C</i> &sub; <i>S</i><sup>1</sup> &times; <i>S</i><sup>1</sup> of <i>C</i><sub>0</sub> is then a loop in the complement of L,<br> and
represents the element (0,1) of &#8484;&times;&#8484; = &pi;<sub>1</sub>(<i>S</i><sup>1</sup> &times; <i>S</i><sup>1</sup>).

Since L lies in the complement of <i>C</i> and L<sub><i>n</i></sub> &rarr; L,<br> there exists an <i>n</i> such that L<sub><i>n</i></sub> lies in the complement of <i>C</i>.<br>
This contradicts your assumtion that p : L<sub><i>n</i></sub> &rarr; <i>S</i><sup>1</sup> is nullhomotopic.