Question about Alexander and Briggs notation

[ComputerGeek]

I am writing a small paper on Knot Theory for my undergrad geometry course and I have a question about the tabulation of knots.

The books I have discuss the different notations, except for the ALexander and Briggs notation, which is used in the tables provided in the appendices.

I believe I understand the notation, but I want to be sure.

$$4_1$$ which refers to the figure eight knot means that this is the first knot of the class of knots that has 4 crossings that cannot be undone by Reidemeister moves. correct? If that is the case, then in the set of knots with 8 fundamental crossings, is there any significance to their ordering in the table? could I take $$8_1$$ and $$8_2$$ and just switch their corresponding knots? also, on some of the knots, there is a superscript. what is the significance of that?

Thanks.

 

[hypermorphism]

In my book (Adams), the superscripts refer to the number of components in the link. I think the subscript is just a historical ordering. For more tables, see the Knot Atlas.

 

[tacman]

Yes, you are correct in your understanding of the Alexander and Briggs notation. The number before the underscore refers to the number of crossings in the knot, and the number after the underscore indicates the specific knot within that class. So, 4_1 is the first knot in the class of knots with 4 crossings.

As for the ordering of knots in the table, there is no specific significance to their ordering. It is simply a way to organize the knots for easier reference. And yes, you could switch the corresponding knots for 8_1 and 8_2 without changing their fundamental properties.

The superscript in some of the knots is used to differentiate between different types of knots within the same class. For example, 8_3 refers to the third knot in the class of knots with 8 crossings, while 8_3^2 refers to a specific type of that knot. The superscript is typically used when there are multiple knots within a class that cannot be transformed into each other through Reidemeister moves.

I hope this helps clarify any confusion you had about the Alexander and Briggs notation. Best of luck with your paper on Knot Theory!