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Question

I am trying to read through the paper Explorations into Knot Theory: Colorability by Rex Butler et al. (2001) and I absolutely cannot grasp the language they are using. I am hopeful that I will provide enough context for anyone trying to answer my question to be able to understand clearly without needing to reference the paper, but it is linked above if it is needed.

Specifically, I am struggling with the $\Leftrightarrow$ symbol, which to me is read in at least 3 different ways between 2 paragraphs. All of these references come from page 8, paragraphs 2 and 3 from the above linked paper.

Firstly we have

$\ldots$ there exists $\beta$ distinct strands such that: to each of those $\beta$ strands any integer from $0$ to $\ell\Leftrightarrow 1$ can be assigned to the strand and there exist numberings (colorings) of the other $n\Leftrightarrow\beta$ strands$\ldots$

I have absolutely no clue what the author means by $\Leftrightarrow$ here and unfortunately I cannot interpret their work without understanding it. Specifically in knot theory, the strands should be assigned values from $0$ to $\ell-1$. These strands are only used in Equation $2$ below, so likely this notation has something to do with the modulo operator, but then $n\Leftrightarrow\beta$ has me at a loss.

Secondly, we have

$$x+y\Leftrightarrow2z=0\tag{1}$$

which we can tell from 3 lines above it in the paper (and which I am also certain on coming in) is equivalent to

$$x+y=2z\mod\ell\tag{2}$$

Before going any further, perhaps Equation $1$ is not precisely the same as Equation $2$ but implies something slightly different. Please let me know if that is the case.

Thirdly, we have

In symbolic notation $K$ is $\ell$ colorable $\Leftrightarrow\exists~\beta~\geq~2~\exists$ distinct stands $\ldots$ such that $\ldots$

The best I can imagine is $\Leftrightarrow$ means if and only if in this context? I also read "$2~\exists$ distinct strands" as "2 such that there exists distinct strands" because that's the only way I'm able to understand it, but I (coming from a physics background, I am not incredibly coherent in symbolic notation) have only really seen $\exists$ to mean "there exists", so perhaps it means something different here?

If you can help I appreciate your assistance. Thank you!

Answer 1

Admittedly, the notation is confusing at first glance. However, I believe that earlier in the passage, this is sufficiently defined as

Each crossing then corresponds to a linear expression given by: $a_i + a_j \Leftrightarrow 2ak$ , where $a_i$ and $a_j$ are the two understrands of the crossing and $a_k$ is the overcrossing.

The $a_i$'s represent colours, and so since the overcrossing is a single strand, they are of the same colour, thus the factor 2. The diagram on page 8 is illustrative of $x+y\Leftrightarrow2z$.

Do you understand how the strands are coloured? I think you should try to clarify that before proceeding. Take a look at how Redeimester moves are done and understand the section in the previous pages. If you understood them, the notation that you are confused about should become clear.

The first two instances are from the definition of the linear expression given above.

Specifically, the first is a reframing the previous sentence

$l-$colorable if and only if the crossing matrix $C_k\mod l$ has nullity $\beta$, where $l,\beta\in\mathbb{N}$ and $\beta\geq2$.

and the second a different characterisation of $C_k$.

Your interpretation of the third instance is correct.