### Update

Due to the pandemic I am currently tutoring online only. I am math tutoring from grades 6-10. In over ten years of tutoring I have had great success.

Due to the pandemic I am currently tutoring online only. I am math tutoring from grades 6-10. In over ten years of tutoring I have had great success.

I am currently working for Oxford Learning in north Richmond Hill. I tutor students in mathematics and physics up to and including the grade 12 level. I also teach privately one-on-one a student from Durham College in Mechanical Engineering Technology. Currently, I am still seeking other students.

I am helping a new student from Wilfred Laurier University. She is taking an online course over the summer which is designed for her to go at her own pace.

I have a new student through ‘ Tutor by Numbers’ he is in grade 12 and is trying to get into Durham College for Automotive Engineering. I have met with him 4 times and will likely be reviewing with him in the summer.

I am working at the Prestige private school located in Richmond Hill. I am tutoring six groups of four students from grade levels 2 to 11 in math.

I am starting a new branch of my business called “Tutor by Numbers”. I am seeking students from grade 4-11 who would be interested in meeting in-person in my local area of Markham/Vaughan. I am specifically looking for but I am not limiting myself to students with learning disabilities.

I am Currently having great success tutoring a student with a learning disability through the Learning Disabilities Association of York Region. He recently received a grade of 100% on two quizzes in math.

Anyone interested in this service can call me at 416-227-1657 or email me at robburchett1@gmail.com.

I am teaching two students, in person, at J.D.L. consulting in Richmond Hill. One student is in grade 9 and the other in grade 10. I also continue teaching a student in grade 8, through the learning disabilities association of York region.

I am still working for JDL consulting in Richmond Hill. I am teaching grade 4 and 5 math. I still continue to tutor a grade 8 student through the learning disabilities association of York Region.

My new student in grade 7 mathematics is doing very well. He received a grade of 88% in algebra last week. Moving forward he is studying fractions.

I have a new student that I started to tutor in math yesterday. I found him through the Learning Disabilities Association of York Region. He is in grade 7.

I am having continued success tutoring grades 8 and 11 at the moment. My students’ grades have improved dramatically.

1. How effective is this tutoring?

A: I have been able to raise grades into the mid nineties

2. Do I have to buy a number of sessions?

A: You do not have to buy any set number of sessions. I can charge you after a set of five sessions or if you prefer, you may pay as you go.

3. What kind of equipment do I need?

A: A basic computer system, high speed internet. You can write to me using a mouse or type your questions to me using a chat window. I am using meeting software. If you have a writing tablet this will work well too.

4. How is payment made?

A :After a face-to-face meeting you can pay me cash as we go. I also accept cheques. You can pay by interact email money transfer as well. For online meetings I am using interact email money transfer.

5.Are you available on demand?

A: No. A specific time to meet is arranged.

The main idea is that two ‘points’ can exist together and still be two ‘points’. Points are as they are usually thought of as that which has location only. ‘Points’ have location but also extended location in a new plane as described below.

On philosophical grounds it should be possible to create an object of two ‘points’ together where we have one point before, since points and ‘points’ both have no extent, this should be possible. (two ‘points’ together should have no extent also, but we could have two items here). Please see the teacup shadow diagram in the post at the bottom. This is not an abstract or imaginary extension of what’s real.

I write my mathematics in plain language to reach the widest audience possible. These ideas do not have to be expressed at the university level, but could be understood much earlier.

We have the concept of the point being an exact location in the plane. Two new ‘points’ can be created together if they are different from the usual idea of points. One may say why two? Why not more? But we can state at the beginning that we have only two. In some other mathematics we may have more.

We can think of two skew lines in three-dimensional space moving together until they intersect. Usually the intersection is thought of as a common point or location. We can also use the phrase ‘two of the same point’. Since if we connect the two lines to form a figure eight, we can go around once and we go over the common point twice.

Suppose we want a different result than a common location. From the philosophy we know we must be able to do this. For this we need to take away the possibility that ‘two of the same point’ leads to one point. In order to have another possibility I have to be able to replace the existing possibility. To do this we need to remove and replace the common point, so we can put something else there. I am not removing a point from a given set of points (the plane), but removing the point itself from the subset of points that make up points of interest from the plane.

This means we need to create a new level of ‘locations’ of a location- where a location is a ‘point’ of interest from a subset of the plane- and ‘locations’ are a new plane of locations which takes over from the first plane where a ‘point’ of interest could be. To take away the concept of the common point I need to add a concept of a location of a point of interest having a location itself.

Then the common point can be removed from the larger set (the subset of points that make up points of interest from the plane can be regarded as a set of points which is contained by the plane of ‘locations’) and replaced and we can create two ‘points’ which do not combine to form a common point-even though they are together. They are two ‘points’ on this new plane. We still have two of the ‘same’ ‘point’. They are the ‘same’ since they have the same location. They are different since they can be moved to any two different ‘locations’ of location.

Needing to be able to take this one point away creates the need and existence of this new level of location. I need to take away the concept of no extent being a point and replace it with a new concept of no extent. This can be an object which has no length, breadth or height but does have parts to it. In this special case the number of parts is two.

I can’t take a ‘point’ away without a new level. I need this space to be able to put two ‘points’ in. (another possibility) If this new plane isn’t here then the two points are coincident and we have just one point. This new plane of ‘locations of locations of points of interest from the plane’ is considered fixed, for now.

These new objects (‘points’) I call ‘parts’ (as in pieces of the whole but in this case the whole is the same size as the parts). Parts are different from points in that I have two items there and not one. They can be labelled the same and are mobile in this new plane. Each doubled location can have two ‘locations’ of location. We can imagine a whole plane of these ‘doublets’. This is the beginning of the work on knots.

We may imagine the empty space in a three-dimensional container. We can extend the point concept stated above to spaces as I can have a set of points be the three-dimensional space. This empty space can be moved to take up the same “space of spaces” as the still spaces, if we move two empty spaces together. Then two empty spaces take up the same “space of spaces” left. . The spaces may be full of something such as the string which makes up a knot. Then the two full spaces can still be at one “space of spaces”. Two items are not together at the same space.

This may be the reality we live in. I am working on some physics in another section of this website. It may be for the far off future, but if we can move space itself then we may be able to travel to one of the exoplanets. Since the speed of light is limited by space but we may not be limited by this speed if we move space itself.

The new level can be what I call placement space. It is a space where a location may have other ‘locations of a location’. Or a small piece of space can have another “space of spaces”. Basically a location or section of space is mobile. At the beginning it is stated how many locations are at any given placement. For the work on knots this is two, and the resulting entity I call a “doublet”. Then two different locations may share the same placement (location of locations) and I call this a “sharing”. A special type of sharing may exist in the manipulation of diagrams which I call a “joining”.

I use the knottedness and chirality or achirality of knots to demonstrate the usefulness of these ideas.

This note is showing the two final diagram reached after the Reidemeister moves in R^3. These diagrams also have to be equivalent in P^3. Then I can set up a congruency of locations between the two diagrams.

This post is describing the push move (named by Murasugi), which is an extra move in knots which doesn’t effect the knot. In R^3, this is seen as the curve being mapped to itself by a rotation of the curve upon itself. In P^3 we see this as a movement of joining pairs around the curve. The cyclic order of the joining pairs gives us the shape of the knot diagram.

This page is showing the different moves possible in the two different spaces. The top space is R^3 and the bottom space is placement space (P^3). Placement space is described in the posts below. The idea that leads to placement space is described in the quick view, below.

There is a link between R^3 and P^3. Also in P^3 we can create a D-sharing- a type of crossing. In turns out in placement space there are two types of basic crossings. I’ve labelled these D and Q. These are shown in some detail in the diagrams below.

Applying moves S1,S2,S3 counterparts to the Reidemeister moves to take us from one diagram to the next. I use Reidemeister moves to go from one diagram to the next, but do not use these to prove knottedness or chirality.

These moves all involve Q-sharings only and I’ve drawn them above. In order that I only apply these moves to Q-sharings a transformation is applied to the trefoil. This transformation is described in the posts below. I’ve shown it in a diagram above.

Then these S1,S2,S3 have counterparts in the space R^3 above. This then leads us to a diagram with the same shape as the original, but with possibly an unknotted case or a mirror image case.

To rule these out we put back in the cyclic permutable set {a(1)r(1),b(2)s(2),a(3)r(3),b(1)s(1),a(2)b(2),b(3)s(3)} which was removed during the transformation in placement space. It might be possible to obtain a mirror image or an unknot because I am applying the moves generally.

That is I allow all possible paths by Reidemeister moves which lead back to the diagram with the same shape. It turns out I can only reform the original configuration, not the unknotted or mirror image case.

For future work I can say that the shape of the knot need not be one shape alone, also in larger knots such as the Perko pair, I can remove the D-sharings and put new D-sharings in, changing the writhe of the knot which is the number of positive crossings minus the number of negative crossings.