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I'm doing research for my thesis and I'm trying to model some type of DNA-associating proteins. I have not yet picked which I would like to work with, but I figured I should give as much background as possible.

I have been trying to understand the basics of self-avoiding walks and self-avoiding polygons but I'm having trouble understanding what $\mathbb{Z^3}$ looks like.

The article I'm reading is "Topology and Geometry of Biopolymers" by E.J. Janse Van Rensburg et. al. It says it easy to see that if $c_n$ is the number of self-avoiding walks with $n$ edges on $\mathbb{Z^3}$ then $c_1=6$, $c_2=30$, $c_4=5\times c_3-24$, etc.

Because I am having trouble understanding the space, this is not easily seen by me. If anyone could explain how to view the space or how they come to this conclusion, I would greatly appreciate it.

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    You could imagine $\mathbb Z^3$ subset of $\mathbb R^3$, that means all points in our 3D space with integer coordinates. Does that help? – 2017-01-11
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    That does help! Thank you! – 2017-01-18

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Fix an origin, we begin with six possible paths, $\pm 1$ in any $x,y,z$ choice. Having reached that first point, there are five directions each that do not point back to the origin, so that makes for 30 paths. At any given stage, we are at a "leaf," a point with one path edge coming in, and five possibilities going out. You do not seem to have typed in $c_3...$

Recommend solving this yourself in dimension two, draw pictures on graph paper https://www.printablepaper.net/category/graph

The numbers of paths will be a bit smaller: $c_1 = 4,$ $c_2 = 12.$

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    This was super helpful, thank you so much for your comment! – 2017-01-18