# Interactive Online Tutoring Services

## December 15, 2022

### Perko Pair demonstration page 1

Filed under: knots,Mathematics,the perko pair — Rob burchett @ 5:44 am

From Knotscape I have found this diagram of Perko A. Crossings 1 through 10 shown (the orientation of the crossings written below the number, all are negative except number 9). Also I found a diagram of Perko B which is shown later. I proceed to move these diagrams into negative space and “undo” them sometimes adding Q-type sharings adding in s and r point(2)’s. This proceeds until I reach a diagram which has the Trefoil as a base for both diagrams. Then I rearrange the joinings to complete the congruency. This is all shown in detail below.

### Perko Pair demonstration page 2

Filed under: knots,Mathematics,the perko pair — Rob burchett @ 5:42 am

Sharing 9 is actually not reduced to (b9(a9) but can be moved to the outside of the trefoil base. There I can put it into congruence with the loop formed from (a17(b17) expressed as (b17(a17).

### Perko Pair demonstration page 3

Filed under: knots,Mathematics,the perko pair — Rob burchett @ 5:41 am

### Perko pair demonstration page 4

Filed under: knots,Mathematics,the perko pair — Rob burchett @ 5:40 am

Here is Perko B, taken from Knotscape. Starting with crossings 11 through 20 (orientations all -ve, shown below crossing numbers). I add Q-sharings and take the diagram apart after moving to negative space as shown in detail below. Then after that I look for a congruency with Perko A, also reduced.

### Perko Pair demonstration page 5

Filed under: knots,Mathematics,the perko pair — Rob burchett @ 5:39 am

### Perko Pair demonstration page 6

Filed under: knots,Mathematics,the perko pair — Rob burchett @ 5:37 am

So sharing 17 is reduced to (a17(b17). In the other reduced diagram sharing 9 is reduced, but we can make it congruent to sharing 17 by expressing (a17(b17) as (b17(a17) and changing the orientation of that crossing. This is where the writhe changes.

### Perko Pair demonstration page 7

Filed under: knots,Mathematics,the perko pair — Rob burchett @ 5:36 am

### Perko Pair demonstration page 8

Filed under: knots,Mathematics,the perko pair — Rob burchett @ 5:35 am

### Perko Pair Demonstration page 9

Filed under: knots,Mathematics,the perko pair — Rob burchett @ 5:33 am