- mathematics of the unknot under tension

The mirror image of S_{1 }È
s_{2}, that is, s_{1 }È
S_{2}, exists by the above lemma. Each motif of s_{1} is in the same position in relation to S_{2} as was the corresponding motif of S_{1} but is of opposite parity. Hence s_{1} is the reflection of S_{1}.

Suppose that the action described above takes place in front of a mirror: as S_{1} is reconstructed on s_{2} to give S_{1 }È
s_{2} an image in the mirror is forming s_{1} (the mirror image of S_{1}) on S_{2} to give s_{1 }È
S_{2}. Similarly, given S_{2} in a loop of string, s_{1} may be added to it. That is, given any string figure, if a two-dimensional subset S_{1} may be unravelled to leave S_{2}, then s_{1} may be formed on S_{2} to give s_{1 }È
S_{2}. Thus any such two-dimensional subset of two-dimensional motifs that can be unravelled can be reformed in the reflected state.

The unravelling and reforming in the same or the reflected state of a subset of motifs is a procedure that affects only the string contacts that constitute the motifs of the subset, not the segments of string left without string contacts by earlier unravellings. Hence the successive unravelled segments of string left by a sequence of unravellings may each in turn, by theorem 1, be reformed in the original or the reflected state (or, in the case of three-dimensional subsets of string contacts consisting of entire motifs together with part or all of the 3D framework of the string figure, in the original state).

Since there are two choices for how each of the U two-dimensional subsets may be reformed, the procedure can lead to any one of 2^{U} look-alikes.

Every look-alike, through some unravelling process, generates a discoverable subset of look-alikes. (Even the trivial string figure generates such a subset, a subset containing 2^{0} = 1 look-alike, namely the trivial string figure itself.) Each discoverable subset so generated includes the generating look-alike: hence the union of the discoverable subsets generated by all look-alikes of a string figure is equal to the set of all look-alikes. In practice a union of a smaller number of discoverable subsets than those generated by all look-alikes will be sufficient.

We assume there is at least one non-empty proper subset which is a shard. Let this shard be called S with catalogue {S_{1}, S_{2}, ... S_{m}}. Let the remainder of the string figure be a member of the set {T_{1}, T_{2}, ... T_{n}} where each T_{j} is a distinct valid state. We shall show that the remainder of the figure is composed of shards.

S is a shard, and so (by the definition of *shard*), for each valid state assumed by the remainder of the figure, S can independently assume any of its valid states. Hence for each T_{j} (that is, for each valid state of the remainder of the figure) we can construct look-alikes {S_{1}, T_{j}}, {S_{2}, T_{j}}, ... {S_{m}, T_{j}}. By listing the look-alikes for each T_{j} we obtain {S_{1}, T_{1}}, {S_{2}, T_{1}}, ... {S_{m}, T_{1}}; {S_{1}, T_{2}}, {S_{2}, T_{2}}, ... {S_{m}, T_{2}}; ... ; {S_{1}, T_{n}}, {S_{2}, T_{n}}, ... {S_{m}, T_{n}}. Rearranging we can list the look-alikes as {S_{1}, T_{1}}, {S_{1}, T_{2}}, ... {S_{1}, T_{n}}; {S_{2}, T_{1}}, {S_{2}, T_{2}}, ... {S_{2}, T_{n}}; ... ; {S_{m}, T_{1}}, {S_{m}, T_{2}}, ... {S_{m}, T_{n}}. Thus, for each S_{i}, T_{j} can assume all valid states. Hence either the remainder of the figure is a shard, or contains a shard. Continuing the argument in this fashion the entire string figure is shown to be composed of shards.