# Interactive Online Tutoring Services

## January 12, 2023

### Trefoil invariant

Filed under: knots,Mathematics,supplemental on knots — Rob burchett @ 3:59 pm

So starting with the standard diagram for the Trefoil knot. I place this in the new space and I have the sharings D1,D2,D3. These have the associated moving parts a1,b1,a2,b2,a3,b3. Then I move the outer loops inward (green and red) forming the sharings Q1,Q2,Q3,Q4,Q5,Q6. These have the associated moving labels r1,s1,r2,s2,r3,s3. Then I form the joinings (a1(r1), (b2(s2), (a3(r3), (b1(s1), (a3(r2), (b3(s3) (all red). Now this can be further decomposed to eventually form the “circle”.

## January 11, 2023

### The third move

Filed under: knots,Mathematics,supplemental on knots — Rob burchett @ 9:18 am

Showing the third move. After reduction, only Q-types are involved.

## January 5, 2023

### Showing necessity to keep orientation of joining

Filed under: knots,Mathematics,supplemental on knots — Rob burchett @ 7:08 pm

Here I show why we must keep to the same orientation when we reform a joining after we undo it in negative space.

## December 31, 2022

### The achirality of 4(1)

Filed under: knots,Mathematics,supplemental on knots — Rob burchett @ 11:42 am

Here is the achirality of the knot 4(1) shown in negative space.

### Showing simplifications

Filed under: knots,Mathematics,supplemental on knots — Rob burchett @ 11:40 am

Here is a better post on simplifications.

## December 15, 2022

### One way of showing knottedness of the Trefoil

Filed under: knots,Mathematics,supplemental on knots — Rob burchett @ 6:53 pm

I place the trefoil, partially back together then move the four joinings through the reformed 2 and 3 sharings until they can possibly form a new diagram. I show it is not possible to form this diagram in positive space so that the trefoil is knotted.