I am wondering if there is a developed theory of "infinite knots" that could capture this object, and tell me something of its knot properties. Imagine vertical helices in $\mathbb{R}^3$, each discrete $(x,y)$ translations of one helix that spirals around the $z$-axis. Here I show translations just in the $x$-direction:
        Helices: 1 Plane http://cs.smith.edu/%7Eorourke/MathOverflow/Helices1Plane.jpg
And here (in lower quality—Sorry!) in both the $x$- and the $y$-directions:
        Helices: 3 Planes http://cs.smith.edu/%7Eorourke/MathOverflow/Helices3Planes.jpg
(I tried to ensure that no two helices touch.) Copying the translations to extend the pattern in $(x,y)$, and extending each helix infinitely in the $\pm z$ direction, seems to lead to a structure that is knotted in some sense. Indeed just two helices seem knotted, although I don't know if this "seems" can be made precise.

Can someone point me to literature on definitions and approaches to such infinite knots? Thanks for your ideas and pointers!

I am wondering if there is a developed theory of "infinite knots" that could capture this object, and tell me something of its knot properties. Imagine vertical helices in $\mathbb{R}^3$, each discrete $(x,y)$ translations of one helix that spirals around the $z$-axis. Here I show translations just in the $x$-direction:
        
And here in both the $x$- and the $y$-directions:
        
(I tried to ensure that no two helices touch.) Copying the translations to extend the pattern in $(x,y)$, and extending each helix infinitely in the $\pm z$ direction, seems to lead to a structure that is knotted in some sense. Indeed just two helices seem knotted, although I don't know if this "seems" can be made precise.

Can someone point me to literature on definitions and approaches to such infinite knots? Thanks for your ideas and pointers!