Interactive Online Tutoring Services

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”.

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

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

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

Here is a better post on simplifications.

So here I am showing a way of seeing if the Trefoil is chiral. I take the 2 and 3 sharings out then see if I can bring them back after going through space in the different way with the 1 sharing. It turns out I can’t do that.

Here I am starting to use other diagrams of a knot. These come about as we are operating in negative space. That is the new space I created with negative distance. In that space I have the ability to pass through, also to move a series of connected parts ( the knot diagram itself) and create joinings (like (a3(b2)). These combined allow me to make new moves.

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.

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.

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).