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About

**Definition**: Given a finite collection of points \(P\) on a circle embedded in 3-space, we say the points \(p,q\) in \(P\) are **consecutive** provided there is an embedded interval in the circle \(I\) such that \(I\cap P\) is \({p,q}\).

**Definition**: A round circle in \(\mathbb R^3\) is a closed curve in an affine plane, having constant curvature. Equivalently, it is the result of intersecting a sphere with an affine plane (a plane that passes through the interior of the sphere, i.e. not disjoint nor kissing).

**Definition**: A knot is an embedded circle in \(\mathbb R^3\).

**Definition**: A circular pentagram on a knot is 5 points on the knot, which also sit on a round circle in \(\mathbb R^3\), such that any two points that are consecutive on the knot are not consecutive on the circle.

This software computes the space of all circular pentagrams on knots, and illustrates it via an animation. We trace out the circular pentagrams as \(p\), one of the five points on the pentagram, completes an orbit around the knot. Generally the circular pentagrams on a knot form a 1-dimensional manifold, and if one fixes a point \(p\) on the knot, there are only finitely many circular pentagrams that pass through \(p\).

The type-2 invariant of knots has many descriptions:

- The coefficient of \(z^2\) in the Conway-normalized Alexander polynomial.
A. Kawauchi,

*A Survey of Knot Theory*, Birkhäuser Basel (2012), https://books.google.ca/books?id=RkEBCAAAQBAJ. - A signed count of the \(X\) chord-diagrams in a planar diagram of the knot. These are called Polyak-Viro formula.
M. Polyak, O. Viro,

*Gauss Diagrams for Formulas of Vassiliev Invariants*, IMRN (1994), No. 11 - As a 'long knot' there is a quadrisecant count.
R. Budney, J. Conant, K. Scannell, D. Sinha,

*New perspectives on self-linking*, Adv. Math.**191**(2005) 78–113. https://doi.org/10.1016/j.aim.2004.03.004. (https://www.sciencedirect.com/science/article/pii/S000187080400074X) - Closely related to (3) is a reinterpretation by Garrett Flowers.
G. Flowers,

*Satanic and thelemic circles on knots*, Journal of Knot Theory and its Ramifications, 22(5) (2013).

In recent work, a variation of these ideas has been used to construct an invariant of the smooth mapping-class group of the 4-manifold \(S^1 \times D^3\), showing that it is not finitely-generated.

R. Budney and D. Gabai, *Knotted 3-Balls in \(S^4\)*, arXiv (2021), https://arxiv.org/abs/1912.09029.

Created by Sean Lee and Ryan Budney.