What part of algebra can I use to describe linked curves? Can anybody tell me what is the algebraic structure of linked curves in 3D space?
Can their linking number be described using a vector space? Or does it require some other structure such as a group, ring ,etc. I am strictly interested in describing the linking number, preferably with a vector space which would allow me to work with matrices. 
For example in the image below, curve 1 is linked once with 2, 3 is linked once with 4 which is linked once with 5. Curve 6 is not linked with any other curve but is self-intersecting having the linking number 3 (if I'm not mistaking). I am interested in describing much more complicated structures than this and I really can't visualize them when curves intermingle each other in complicated ways.

I would like a little guidance, as I can do the grunt work myself. If possible, an online reference recommendation with a gentle introduction to the subject would much appreciated.

Racks and quandles are associated with knot theory. In fact, it does not make any sense at all to mention linking number without talking about knot theory. 
You have checked out knot theory, right?  There are invariants of knots that you can find, and racks and quandles are supposed to be used in relation to them.
