There is something "in" the object we call a "knot" per se, which contributes to the easiness we can handle it mentally, in our mind s 2D "screen" : the representation and the transformation of knots becomes very simple and unambiguous, when the knot is spread out in a 2D real or imaginary "flat surface", and all crossing points are crossing points of two, and two only, lines. The same happens in mathematical knots : To represent their knots, mathematicians use "flat" diagrams, where all crossing points correspond to two, and two only, overlapping lines at each point, although their "lines" are supposed to be infinitely stretchable, and they could had used any "curved" and tangled representation they had wished.

So, it seems that there is something "in the knots", or, at least, "in our mind", which makes us represent the loose knots on a 2D surface, that is, as been "flattened out". Of course, a finished 3D knot can be loosened and then squeezed on any 2D flat surface we wish, there is no "preferable", intrinsically determined orientation to do this. However, there will always be one orientation from which a projection of the compact 3D knot on a 2D surface will have a maximal area cross section, and one from which such a projection will have a minimal area cross section. From this one orientation the knot will look more "flat", and from the other one it will look more "globular". We chose to select the former and call the view from it "front" or "rear" , and prefer to represent the loosened or the compact knots in this orientation, simply because when the cross section becomes larger, the distance between any two crossing points become larger, too, and the possibility to have "ambiguous" crossing points with three or more lines overlapping each other becomes smaller.

This means that our choice of the "flat" "front/back" view is not as arbitrary as one might think it is... There

*are* some knots where, initially, it is arbitrary, indeed, when the knot is axially symmetric - as the beautiful

*Diamond* bend and PET loop is, for example :

http://igkt.net/sm/index.php?topic=5151.0 However, if we chose to project the pair of the Standing/Tail Ends so that the two lines do not overlap each other, they are not "over" or "under" each other, but "side by side", the one next to the other, we get a preferred orientation for those knots, too ( see the attached picture ).

http://igkt.net/sm/index.php?topic=5151.msg33744#msg33744 In short, all knots, in their loose or their most compact form(s), can be represented on one more or less preferable "flat" surface, and the fact that such a surface may not be actually/physically available, "at hand", does not mean that it can not be mentally available, in "the mind s eye". It is this surface which helps us remember and manipulate the knots, more than anything else. When a climber ties and unties a knot, he should be able to do this when he wears gloves, and his ropes and fingers are almost frozen, and in the dark, and hung upside down, and behind his back !

There is no flat surface anywhere close - but there is always a flat representation of those knots in his mind, without which he can do nothing at all. See all representations of the fig.8 knot : they are flat, and moreover they are flattened by been spread out on a surface of always the same orientation, although the fig.7 knot is a complex convoluted 3D shape, which could had been viewed from any angle. Do you always carry this 2D surface in your brain s luggage ? Then do not be afraid of the absence of the table you forgot at home...