V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 1 sur 59 ©Charles HAMEL aka Nautile 4 BENDS : Zeppelin, Hunter , Ashley, Alpine Butterfly A COMPARATIVE STUDY. 23 pages of text the rest, 36 pages, is illustration : photo, figures, tables. SUMMARY page 2 OBJECTIVE O F THE STUDY page 2 DIFFERENT TYPES OF BENDS page 3 THE BENDS Page 4 EXISTING SITUATION page 5 INTRODUCTORY REMARKS page 6 MATERIAL AND METHODS page 11 DATA COLLECTING page 33 Zeppelin vs Hunter page 33 Ashley#1452 vs Alpine Butterfly page 35 CONCLUSION page 38 V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 2 sur 59 ©Charles HAMEL aka Nautile SUMMARY: Four bends ( Zeppelin, Hunter’s, Ashley #1452, Alpine Butterfly) are examined on their sole morphology and about an alleged “topological similarity”. There is no intent of evaluating the comparative practical merits or lack of therein. Beyond the simplistic similarity that can be stated as : they are bends, the joining of two ropes, in these cases using two knots to create a locking mechanism, NO NON- TRIVIAL SIMILARITY was found. OBJECTIVE OF THE STUDY There is absolutely no intent of evaluating “practical merits or lack of therein” but only the legend of similarity (topological or not). The author holds the view that the use of the noun ‘similarity’ in relation with any pair in this “gang of four” is an abuse of guarded and educated language. (similar == identical but for a difference in size) The author’s contention is that only a low quality observation can find any medium or high order similarity beyond the very trivial one : All are comprised of two lengths of rope made into a functional single length using a joining structure ( a bend ) consisting of two interlocking knots made with the WEnd of the ropes. Each WEnd makes an individual knot for its own AND participates in its interlocking with the knot of the other side. One of the two original SParts keeps its SPart nature while the other mutate into the WEnd of the newly made single length of rope. The interlocking knots are thumb knots ( what is called “ “simple knot” in French because it is the smallest knot that can exist ; no knot can exist under 3 crossings) that can take 4 forms: “S” OVER hand, “Z” OVER hand, “S” UNDER hand “Z” UNDER hand ( see next page about some of the types of bends ) V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 3 sur 59 ©Charles HAMEL aka Nautile DIFFERENT TYPE OF BENDS *** the two WEnd take their own part in the making of the joining knot- Opposite side tails (#1406) *** each WEnd makes its own knot and also contributes to its interlocking with the opposite knot. Same side tails (#1408 #1425 #1453 are two thumb knots) *** the two WEnd follow a unique cordage route to form one joining knot. Same side tails. (#1410 *** each WEnd makes its own knot and also contributes to its interlocking with the opposite knot. Opposite side tails (#1412 #1426 are two thumb knots) *** the two WEnd form each a knot which is not interlocked with the opposite knots as the bend is formed by the two knots abutting to each other and so locking the bend. Opposite side tail (#1414) The two WEnd form each a complete knot which is not interlocking with the opposite knot but the SPart of the opposite knot pass through it and so the two knots are abutting to each other and locking the bend (#1416) *** the two WEnd each form a part of a common joining knot. Same side tails perpendicular) (#1418) *** the two WEnd do not form a real knot ( 3 crossing minimum) but each makes a half-hitch to which the SPart of the opposite half-hitch give a “support” in cordage and the two half- hitch abut against each other so locking the joining. Opposite side tails ( #1420) *** the two WEnd each form a half-hitch which interlocks forming a knot. Same side tails (#1422) *** just a intertwining of the two WEnd. Opposite side tails (#1423) *** the two WEnd each form a half-hitch. The half hitches are interlocking to form the joining knot. Opposite side tail ( #1428) *** each WEnd form a loop and loops are interlocking. (#1455) Those are some of the types. It is obvious that one type cannot be similar to another type or they would no be really individualized ‘types’ but a single type. The question that remains is the one about the similarity between bends belonging to the same type. V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 4 sur 59 ©Charles HAMEL aka Nautile THE BENDS Photo 1 & 2 THE BENDS MADE WITH COMPARABILITY IN-BUILT AND MAINTAINED ( as explained latter) The Roger Tory Peterson ‘s system (ornithology) is used to point the discriminators. V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 5 sur 59 ©Charles HAMEL aka Nautile EXISTING SITUATION There exists quite a lot of parroting about “ topological similarity ”. 181 entries in Google with "It is topologically similar to the Zeppelin bend" in the search field on 2011 Oct 21th. Wikipedia [open quote] The Hunter's bend (or Rigger's bend) is a knot used to join two lines. It consists of interlocking overhand knots, and can jam under moderate strain. It is topologically similar to the Zeppelin bend. [end quote ] This is the single one I kept as “sample”, eight samplings I deleted : it is indeed just boring reading as it is only tedious and unimaginative as all plagiary is. ( a heartfelt thank you to Constant XARAX for pointing to me that non information redundancy ) There exists also another kind of plagiary who wisely refrains from using “topology” ; only seven samplings of those were deleted for the same reasons the eight previous ones were deleted. [open quote]… zeppelin and Hunter's Bend interesting rigging knots that are very similar [end quote] When it happened that those persons thought to precise the nature of the thumb knots they speak of “OVERhand”, never of UNDERhand, and never say a word about an important topological parameter : chirality. V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 6 sur 59 ©Charles HAMEL aka Nautile INTRODUCTORY REMARKS The THUMB KNOT can take several forms, it can shows four different aspects of itself to the observer if that observer is forbidden to manipulate the knot from its “position” directly resulting from the making of the knot. Ashley is showing #46 which is by structure a ‘Z’ UNDER HAND while saying it is an OVER HAND. The thumb knot must be distinguished in Flat Land, (2D) diagrams, not only as OVER hand and UNDER hand but since each of the two can take two chiral forms we also get : “S” OVER hand, “Z” OVER hand, “S” UNDER hand “Z” UNDER hand. There is no possible similarity between a “Z” and a “ S” as they are plainly different at first sight. OF COURSE , in the cordage as opposed to ‘in the diagram”, as you have no information on the “spatial position” in which the structure was made, there is a collapsing of details and there are only two ‘types” of orientation to take in account : “ S” and “ Z”. As shown in Fig 0 bis page 14 the CW OVER “ S” is equivalent (Reidemeister’s moves) to the CCW “ S” ( CW UNDER “ Z” to CCW OVER “ Z”). This is so because once the knot is made we lose, by manipulating it, very important information about its orientation in the making phase, information which is kept in the V1.0 2011 Oct 21th V 3..0.27 2013 March Public Release version V1.0 Page 7 sur 59 ©Charles HAMEL aka Nautile diagrams representations which show beyond any reasonable doubt that there are FOUR “forms” of formation processes of the Thumb knot. Four ‘ forms’ that by not being cognizant or by deliberately ignoring or by destroying important information about the actual making can be pragmatically collapsed into the TWO types “ Z” and “ S”. The use of left /right hand(edness) about any entity devoid of a brain is egregiously improper as handedness is a neuro-psychological preference. Better say ‘ left /right oriented’ so that one can avoid to commit the illogical thinking : "left -handed" (right -handed) because it was made left -handed ( right -handed). The left /right orientation notion does not pertain to the " maker' nor to the ‘making’ but to the " made " and simple rules taken from topology allow to consistently determine the orientation of the knot, which has absolutely nothing to do with the orientation of the " maker " or of "the making ". When with a ‘in the cordage’ “ S” thumb knot *made* CW OVER you show it as having the appearance of an “ S” thumb knot *made* CCW UNDER it is *only* at the cost of introducing a very important ‘existential change” in the knot : you need to give it some energy to make a flip (turning it like a book page). It is identical to the other member of the pair only by ignoring this important fact : a flip was introduced in one and not in the other which *prove* that the making was quite different and sustain the FOUR “forms” for TWO “types” “ Z” and “ S”. Suppose that it is forbidden to flip a knot in the cordage from its ‘making’ position without putting a paint mark on it "to keep the existential memory" : then you will have indeed two quite distinguishable 'forms' having been made differently.