The topology of a knot does not uniquely determine its geometry. *Topology does determine zero to several allowable *knotted* geometrical conformations - in other words it determines a *set* of materially *possible* geometrical configurations * -so it does not lead always to one and one only knot.

So, one and one only tying diagram can lead to two topologically identical, but geometrically quite different knots. In this thread one can see two such knots, a Pretzel-to-Pretzel bend ( where each of the two interweaved links of the bend is a Pretzel-shaped overhand knot, shown at the attached pictures of this post ) and the Hunter s X bend ( the well known Hunter s bend, where the tails are crossed before they leave the knot s nub, shown at the attached pictures of the next post).

Are those knots two different knots ? Yes, I believe they are. Why ? First, because they *look* so different... as everybody can easily see by just looking at those pictures. Moreover, and this is the most important thing, they are different because in each of those two knots the standing parts follow very different paths in space, and so they are loaded very differently. If we had been able to "see" the flow of the tensile forces as they "run" along their carriers, the standing parts ( going from the 100% of the load at the standing end to the 0% of the load at the tail), there would have been no doubt whatsoever about this.

Why does the one and the one only tying diagram lead to two different knots ? It seems that, starting from an initial configuration described by a particular tying diagram of a knot, and proceeding by pulling the standing ends, we can reach one and one only stable form of this knot- and not any other. As the knot shrinks, and its volume is reduced, it will arrive at some level of stability, and it will settle to one stable compact form, from which any further pull of its standing ends will not be able to change its geometry. To transform this one stable form of the knot to another one, one has to intervene manually, and pull or rotate the collars of the knot, that is, disturb the achieved balance. Then, starting from a different initial configuration, the knot will be able to reach another lever of stability, and it will settle to a second, different stable form.

"Pull or rotate the collars of the knot". I have done something like this to the Pretzel-to-Pretzel bend, shown in this post, which is a very stable, compact and secure knot, and arrived at the Hunter s X bend, shown at the next post. So, those two knots are two stable different forms of one parent "bistable" knot - or two different stable knots.