The 9-1 Knot

Knots and crossing numbers

A knot is a simple, closed, non-self-intersecting curve in \(R^3\). It is natural to think of a knot as constructed from a string glued together at the ends, usually tangled in the middle.
The knot for my project is knot 9-1 in the knot atlas, which is centrosymmetric. The crossing number of the knot is 9.

It is common to use a knot diagram or 2D projection to graphically represent knots. A knot is represented by a curved line on the page. When the knot passes under itself, a gap is used at the crossing. The projection of the 9-1 knot is shown in the following picture:

Sequence of crossing switches for unknotting

There are three types of simple allowed ways to deform knot diagrams by changing the number of crossings as shown in the following picture.

The unknotting number of a knot is the minimum number of times to switch the crossing points to untie it. The 9-1 knot has the unknotting number \(4\). As shown in the following picture, the sequence of crossings \(1,3,5,7\) marked by red dashed line circles can be switched, then by using the Type II and Type I deforming methods, the knot is untied.

Coloring the knot

The 9-1 knot is colorable under the following rules for coloring a crossing

all three colors are used; or
only one color is used.

The tricolor effect of the 9-1 knot is shown in the following figure.

The writhe of the knot

Knots are frequently considered with orientation. That is, the strand of the knot is given one of two directions, indicated by arrows along the strand. Crossings can be either right-handed crossings (positive crossing, denoted by \(+1\) ), or left-handed crossings (negative crossing, denoted by \(-1\) ) with respect to the current orientation of a projection. In a knot with one component, reversing the orientation does not alter the handedness of its crossings.

The orientation of the 9-1 knot is shown in the following figure.

The writhe of a knot is the total number of positive crossings minus the total number of negative crossings. There are 9 positive crossings in total for the 9-1 knot. The writhe of the knot is \(+9\).

Polynomial Q

The knot polynomial is a kind of knot invariants in the form of a polynomial. The Polynomial Q is calculated by splitting crossings according to the following basic rule.

The first two steps of the polynomial Q of the 9-1 knot is shown in the following picture.

In this project, I investigated the characteristics of the 9-1 knot. The final 3D print was interesting and beautiful.

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