General post

Enumerating simple interlocking overhand bends

Originally posted 2024-12-23

Last updated 2024-12-24

Background

This is to serve as context for an upcoming meetup where we’ll discuss the closely related bends which consist of “simple” interlocking overhand knots.

This page enumerates all possible bends which can be formed by interlocking two single overhand knots in a “simple” way. A “simple” construction works as follows:

  1. Form a three-quarter turn with the “left” rope. The standing end should be coming in from the left side and the working end should finish “upward”. The partial turn should essentially look like a lowercase “b”, with the helical part having a left-chirality (working end sits on top of the standing part, as shown in the diagrams).
  2. Form a three-quarter turn with the “right” rope. The standing end should be coming in from the right side (since the joined ropes will form a 180-degree angle). In contrast to the left turn, the right one should be either “d”- or “q”-shaped (depending on which way the tails exit). The right helix may be either right-handed or left-handed.
  3. The right and left loops may be interlocked such that they cross at most once. That is, the right can be stacked underneath or above the left loop, or else it may enter the loop on top (and exit below) or enter the loop from below (and exit on top). The loops cannot be interwoven beyond this.
  4. Finally, the working ends should be passed through the shared loop area formed by the stacked or interwoven loops. There is only one possible way to pass the working ends such that the respective overhands are completed.

There are obviously infinitely many variations on the above if you allow other interleavings or tail work, but there are only 16 “simple” bends as described above.1 Not all of them hold fast; I have indicated which do and do not below. All well-known variants do hold fast, and some others do as well. However, my bar for “holding fast” is pretty loose and should not be taken as an endorsement of a not for any real workload; it’s simply an observation that a knot does not immediately come undone as soon as tension is applied.

For knots that are strikingly similar to well known knots (but are otherwise unnamed as far as I’m aware), I have added a “false” or “anti-” prefix to their names. I have used “false” to describe a knot which deviates from its well known variant in such a way that it fundamentally breaks the knot’s security. I have used “anti-” to describe a knot that has some reverse chirality or other structural “oppositeness” which identifiably changes the knot but does not necessarily break security.

Finally, how the knots are dressed has a material impact on the knot structure in some cases. This is particularly important in all the “false butterfly” variants, which end up as offset fisherman’s bends if not “interlocked” as intended.2

Yes, I’ve named a knot after myself.3

Anti-Hunter's Bend
Similar to the hunter's bend, but the incoming legs are swapped relative to the base overhands. This results in extra torque on the bend, causing it to collapse (but not quite capsize) under tension, unlike the true hunter's bend. Pleasing and symmetric. The tail-swapped hunter turns out to be a non-"simple" bend, so it is not shown here.
Shake-Hands Bend
Secure, but not as apparently as inherently secure as the other well known interlocked overhands due to only the outer lobes being interlocked. Bulky and takes some care to dress well, but symmetric.
Hunter's Bend
Very secure but jamming. (Under my tests, I was able to easily "break its back", but the tail-swapped (non-simple) version is a beast to untie after hand loading.) Dresses nicely and is symmetric. Well known.
False Fisherman's Bend
This is essentially a single fisherman's bend with the respective over hands tied on the wrong side of each other.
False Butterfly Bend A
Appears similar to the butterfly bend at first glance, but not interlocked securely. The overhands don't properly clamp onto each other and loosen under tension (but not as bad as the false fisherman.) Note that the dressing matters. In the "false butterly" configuration, the overhands should be interlocked. If you allow the orange knot to slide past the blue knot when dressing, you end up with an offset fisherman's bend (opposite chiralities).
Butterfly Bend A
Secure and non-jamming. Well known and widely used.
Butterfly Bend B
Variation of butterfly A with different crossings but the same overhand chiralities.
False Butterfly Bend B
Variation of false butterfly A. In this case, letting the blue knot slip past the orange knot results in an offset fisherman's bend (opposite chiralities).
False Anti-Butterfly A
Looks like the anti-butterfly (wings opposed), but not interlocked securely. If you allow the orange knot to slip through the blue knot, you get an offset fisherman's bend (matching chirality).
Anti-Butterfly A
Looks like the butterfly but its wings are opposed.
Ashley Bend
Pleasing and secure but bulky. Can be distinguished from the butterfly because the wings want to sit perpendicular to the normal direction as on a well-dressed butterfly. The interlocking forms lots of extra pressure on the exiting tails. Jamming. Well known.
False Anti-Butterfly B
Variation of false anti-butterfly A. If you let the blue knot slip through the orange knot when dressing, you get an offset fisherman's bend (matching chirality).
Zeppelin Bend
Pleasing, symmetric, strong, and non-jamming. Easy to visually verify. Bulky. Well known. Like the hunter's bend, the tail-swapped variant of the zeppelin is not a "simple" knot and jams easily.
Ben's Ugly Bend
It's not symmetric or neat looking; difficult to dress properly, but appears to be stable when it is "elongated" along the axis of pull of the standing ends. It's particularly displeasing because one of the lobes of the left (blue) overhand ends up capturing 3 rope diameters (2 orange, 1 blue), while the orange part only captures 2 (1 orange, 1 blue). The remaining blue crossing is instead separated by the other orange lobe.
Offset Shake-Hands Bend
A bit finicky to dress properly, but appears to be stable when it is "elongated" along the axis of pull of the standing ends. Tails extend on opposite sides of the bend, at an angle similar to the shake-hands. Not well known.
False Anti-Fisherman's Bend
Compare to the false fisherman's bend. This is the same situation, but with the respective overhands having opposing chiralities. While still inherently insecure, this knot holds much stronger than the false fisherman's knot due to how the opposing overhands hold each others' tails. Interestingly, this appears to be in contrast to the proper fisherman's bend, which is more stable when the overhands (single or double) are tied with matching chiralities.

Footnotes

  1. I’ve eliminated duplicates due to reflections or simple rotations by fixing the chirality and orientation of the left loop and assuming the ropes are identifiable (here marked in orange or blue). All variants shown here are “distinct” in that they are formed by different crossings and have distinguishable structure, even if they behave equivalently under load. I have labeled “A” and “B” variants in such cases. ↩

  2. I plan to eventually upload photos of all completed knots as they should appear after proper dressing. ↩

  3. This knot and all the “false” and “anti-” knots were new to me. I will update if I learn of established names for any of these. ↩

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