Batch 2 - Class 313 - Topology - Knots, Celtic Knots

• Understand if students have written code for wordle and how well it is working
• Find the right angled triangle with the largest perimeter, with one of the sides measuring 13 units

• Where have students experienced knots?
• Knots appear in topology studies - a circular string is the simplest knot. The next one is trefoil, which is a knotted doughnut

• It looks impossible in 2D, but we can visualize in 3D. In 4D, this can be untangled, so it is not even a knot!
• Topological equivalence is not an intuitive problem. To determine whether two surfaces or knots are equivalent is a hard problem. Why is this problem interesting:
• For example, DNA can often get tangled. Theory of knots can help us understand those patterns and figure out efficient ways of "unknotting" the DNA. There are other applications in cryptography and quantum computing

• If you hanged a picture using a nail, and then removed the nail, what would happen?
• If you hanged a picture using two nails, and then removed the nail, what would happen?
• Think again, it is necessary that it would continue to hang?
• Can you think about a way of hanging it, where if you remove one nail, then the picture falls?
• How do we think about it? Lets start from normal way of hanging a picture on two nails, and then start to take one nail out of the picture at a time. See https://www.think-maths.co.uk/picture-hanging
• Alternately, one can think about balancing turns around each hook with equal number of clockwise and anti-clockwise loops.
• What if there were three hooks?
• Lets students solve, and then see if we can go from a solution for n nails to (n+1)
• Alternate solution around binary thinking and every clockwise canceling with anticlockwise. Inserting a new nail now just means going clockwise and anticlockwise around the new nail, but so that if we remove that new nail, then rest of the structure collapses.
• How does this scale - what is the number of turns we require for n nails - its exponential. Can you think about a way to make it more efficient? The principle is same as binary search - rather than divide the solution for n into solution for (n-1) and 1, we divide it into two solutions of (n/2)

• Here's a video on how mathematicians do it https://www.youtube.com/watch?v=XPIgR89jv3Q

• What if you wanted the picture to fall if you remove 2 out of n nails? (See the advanced paper in links below). Solve for 3 nails