The first ordered fixed loop is revealed to be the Thief Knot

New Knot Investigations
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The first ordered fixed loop is revealed to be the Thief Knot.
The first ordered non-trivial single-line knot form is the Trefoil Knot form, a form of three knot crossings. The Trefoil is the model for a number of knots including the Overhand Stopper Knot, the Overhand Bend, and the Noose. The Trefoil can be the pattern of a fixed loop when tying the Water Knot as a loop or when tying the Overhand Loop. The Water Knot Loop is nearly two complete circuits of the Trefoil form with a total of six knot crossings. The Overhand Loop is two nearly complete circuits of the Trefoil form with a total of six knot crossings.
The second ordered single-line knot form is the Figure Eight Knot form, a form with four knot crossings. The Figure Eight Knot form is the pattern for a number of practical knots including the Retraced Figure Eight Loop with its two nearly complete circuits and eight total knot crossings.
A loop knot modeled on the pattern of the Thief Knot will follow the circuitry of the Figure Eight Knot form. The circuit is completed with the working end making a fifth knot crossing. Such a loop knot will have some stability when sufficiently tightened in a material with sufficient friction. This fixed loop, the Thief Loop, is the first ordered for having just five knot crossings, a minimal number.

Trefoil and Figure Eight Knot forms.jpg

Water Knot Loop, Overhand Loop, and Retraced Figure Eight Loop.jpg

Thief Loop tying diagram.jpg

Thief Loop tied.jpg

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A too “minimal” number : it has six X-ings (2x3-each-half).

(There’s an interesting redressing of the Thief that much
captures/has Fig.8 geometry, and ten X-ings! And a securing
of the Grief looking impressive. --things arising from my
musing about a recent article drawn to my attention, viz. :
Tangled Physics: Knots Strain Intuitive Physical Reasoning,
by Sholei Croom and Chaz Firestone, inOpen Mind, Vol. 8; September 2024
)

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