Let two d-dimensional manifolds M and N. By removing an n-dimensional ball from each of them and identifying the resulting boundaries, one obtains a new manifold which is called the connected sum M # N .

Prime Manifolds

A manifold that cannot be decomposed as a connected sum of other manifolds, is called a prime manifold . One should exclude from such decomposition the connected sum of the original manifold and the n-spheres.

d=1, Knot Theory Application

According to the procedure stated above, one may start from a number of non-trivial knots and obtain their connected sum. To do so, one cuts each of the knots at some point and thus each knot obtains two boundary points. Then one glues boundary points of different knots. This gluing should fulfill the following rule.

There must exist one (at least) sphere dividing the 3-dimensional space into two disjoint segments, such that each of the two knots to be summed belongs entirely to a different segment, with the exception of the boundary points that belong to the dividing sphere.

In a similar way one may perform the connected sum of knot shadows. When one attempts to draw a notation, one faces a chirality ambiguity each time one reaches the first crossing point of a new prime constituent of the knot shadow. To give an example, starting from {(1,4),(3,6),(5,2)} one may draw either the right or the left-handed trefoil. This ambiguity is resolved as follows.

First, one disregards knots that are connected sums. Such knots can easily be detected. There is always some subset S of {1,2,...,2N} other than the empty set and {1,2,...,2N} such that for all (i,j) belonging to the least crossings notation, i belongs to S <=> j belongs to S. If one is really interested in obtaining such knots as well, one may combine prime knots, and thus the elimination is harmless. One is thus left with merely two possibilities related by mirror symmetry. Second, one identifies the two mirror images. This is not technically correct. (Similarly, when we allowed for orientation reversal in a previous page, that too violated the rules of the game) At this point however we have to allow for such a violation; we shall explain this statement by using the trefoil as an example.

Through recently invented knot characteristics one may easily show that the trefoil and its mirror image are not ambient isotopic. This cannot be shown however through the color tests, since both trefoils have the same group. Were we to insist on regarding the two trefoils distinct, we would be unable to join them through Reidemeister moves, and to distinguish them through color tests, and thus knot classification would fail from the very first non-trivial knot. We thus opted to consider mirror symmetric knots equivalent, and leave the detailed discussion for another moment.

By considering mirror symmetric knots equivalent, one considers the notations {(i,j)} and {(j,i)} identical. According to the criterion established in a previous page, one thus considers only notations where number 1 appears at the left of its pair and is thus an overcrossing.

At this point the classification of regular knot projections is complete. Click here to proceed with the discussion.

Charilaos Aneziris,

Copyright 1995


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