Knot 9-47

Knot, we have been in contact with it from a very young age. Perhaps the most relevant problem encountered is tying shoelaces. Sometimes I get annoyed by the mess of earphones in my pocket (sometimes it is really difficult to untie the knot). As the last blog of this semester, we want to discuss this object that we can encounter in many places in life, but perhaps we have never studied seriously: knot.

My model is finished long ago. It belongs to 9-47 and is a knot with 9 crossings. Then let's study some of its properties.

crossing number:

As I said before, its crossing number is 9. It looks like this:

Unknot:

We now try to unknot this knot, in other words, open the whole knot by transposing the knot in it, we try to unknotted so that the whole knot becomes a ring, this is our purpose.After my attempts, I found that it can be unlocked up to three times:

But for 10 crossing knots, there have been the least number of answers: 2. Obviously this is not the easiest way, so I have tried many times and came up with the simplest way to unknot these two points：

So in the end, according to my many attempts, the knot can be solved at least twice, and the maximum should be 3 times (multiple attempts can be solved in three times)

Tricolorability:

This should belong to one of the attributes of knots. Some can achieve this attribute, but some cannot. A knot is tricolorable if each strand of the knot diagram can be colored one of three colors, subject to the following rules:

1. At least two colors must be used, and

2. At each crossing, the three incident strands are either all the same color or all different colors.

Then every node needs to be detected, either three colors intersect, or only one color can appear.

As you can see from this animation, I'm lucky that the knot I chose has tricolorability. It can be colored in three colors according to the law.

writhe

It is a property of the knot. We divide the crossing in the knot into two types and assign them real numbers (+1, -1), so the writhe of the knot is the total number of all the knots, so what is + What is the case of 1 as -1? It follows the following principles:

It can be seen that the curve from right to left ane its one the top: +1, otherwise it is -1. Then we give the knot a direction, we can get:

Then according to the figure, the blue arrow is the direction, so that each intersection will produce a value, the total of +1 is three times, -1 is six times, so the writhe is -3. -3 This result will not change due to the change in the direction of blue, but all crossings can be exchanged, in this case the writhe will be +3.

Q Polynomial Steps:

This part is very complicated, but it is not because of the difficulty of this part, but because it will be more cumbersome, so only the first few steps are shown here.

Then the picture above is the rule of untie knots. Due to its complexity, only two of them will be unwrapped this time.

I chose this knot from many knots, mainly because it looks symmetrical, and the distribution of all crossings seems to be relatively even. After further study, I found that there are so many attributes for the knot. Fortunately, the knot I chose meets Tricolorability, because it looks amazing, which makes me love this knot even more.