Treebeard's Stumper Answer

Kinky Chains
While tuning up my bike for the Dunn Middle School Refugio Ridge Ride on Friday, my chain slipped off the rear sprocket and ended up tangled in the peculiar kinked loop shown below. Bike chains are stiff, so you can't just turn a loop over, and pulling on it only makes it worse. Since I didn't actually remove any link to get into this fix, I know it must be possible to get my chain unkinked. The stumper is to clearly visualize step by step how to create kinky chain loops and how to undo them. Can you find similar patterns that can't be made or undone without removing a link?
Here's a photo of the kinks in my bike chain, and an illustration that shows the overs and unders. To really appreciate this stumper, you should play with a real bike chain and try to make and undo the knots. (Wear rubber gloves unless the chain is new!) The circle on the left represents the front chainwheel which traps the chain. It's interesting that when I finally untangled my chain, I didn't know how I did it! The stumper is to visualize the process step by step.
Many DMS students have become experts at making kinky bike chain loops and undoing them. It can be done with a single motion, but it's hard to explain how we do it! Here is my best attempt at visualizing the process step by step. I think there are (at least) three kinds of bike chain knots. This stumper can be undone while the chain is trapped by the front sprocket. Others can only be undone if the chain is free. Some can only be undone with a "chainpuller" tool to actually open the chain. I have examples below of each kind and some analysis.
Notes:
I found this stumper in some published lecture notes on Geometry and the Imagination by John Conway, Peter Doyle, Jane Gilman, and Bill Thurston. It awakened real memories. This has happened to me several times with my bike and chainsaw, and I was thoroughly confused every time. I always knew it must be possible to unravel the kinky chain, and I always managed to do it after a while without quite knowing what I did. Now I understand.
Mathematician John Conway says that "Geometric imagery is not just something that either you are born with or you are not. Like any other skill, it is something that needs to be developed with practice." Showing my solution step by step was excellent practice. See if you can visualize this as a movie forwards and backwards in your imagination. If you find the opportunity to unravel a real bike chain on the road, then you will impress everyone!
I figure there are these three different kinds of kinky chain knots.
 Type 1  Chain knots that can be made and undone while the chain is trapped.
 For example, the kinky chain in this stumper which is solved by sliding one loop over the other. All the action is on one end of the chain loop, so the other end can stay trapped.
 Type 2  Chain knots that can be made and undone only if the chain is not trapped.
 For example, this very similar knot can be easily undone if the top strand is pulled down below the bottom. But that's impossible if the chain is constrained by a front gear sprocket.
 Type 3  Chain knots that can only be made and undone if the original chain is opened and then reassembled.
 For example, this simple loop cannot be made or undone without using a chainpuller to break the chain and then reassemble it when done.
I'm way too busy with my upcoming Dunn Middle School Science Fair, Slide Show, and Graduation Trip right now to do a real analysis of kinky chain loops, though I'm sure an elegant analysis is possible. As a middle school science teacher, I'm always proud to say "I don't know!"
Graybear is busy too:
I didn't have time this week to get too far into it, I wanted to come up with a universal way to look at a chain and be able to tell whether it could be returned to the original shape without removing the masterlink. All I've figured out so far is that if you color code the outer side of the chain say, blue, and the inner side of the chain, say, red, then the number of loops with blue on the outside minus the number of loops with red on the outside must equal one.I'm sure the numbers of overs and unders are also important. So here's a real stumper as we move into summer, with a good hint from Graybear. I don't know the answer, but here are a few more kinky chains to ponder. Are these type 1, 2, or 3?
Here are some more links for your own research:
 I haven't found anything about this stumper on the Web except for the original lecture notes on Geometry and the Imagination by John Conway, Peter Doyle, Jane Gilman, and Bill Thurston. These notes are also available as a PDF file which prints well. There's an account of John Conway's teaching style in giving these lectures. These lecture notes are a rich source that I also mined for my Bicycle Tracks (9 March 2001) stumper. I'll use them again!
 There's info in Conway's notes about the mathematics of Knot Theory, but bike chains are different since these chain loops cannot be flipped over though they can slide past each other. These chain knots aren't quite 2 dimensional or 3 dimensional! There's more info on Knot Theory from Los Alamos National Laboratory, The Knotplot Site (The University of British Columbia), York University, and The Geometry Junkyard. I'm sure there are many more sites if you do a Web search.
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