| In the opposite way,
from a Gauss code we can simply read the corresponding graph. Its corresponding
list of unordered pairs can be read directly from the Gauss code: every
unordered pair of adjacent numbers in each component (where the first and
the last number in every component are adjacent as well) gives one edge.
A visual drawing instruction probably doesn't need further
explanation.
The Gauss code of a KL is very flexible: it is invariant with regard to a change of the order of components, to a rotation of any component, or even to its reversing (if we are not interested in the orientation of components). This follows directly from the fact that every time, beginning to trace a new component, we choose the beginning point and the orientation- the first oriented edge going from the beginning point. By Component Algorithm, in tracing circuits for every circuit we choose a beginning point, and a first edge that induces the orientation of a whole circuit. Hence, even for knot shadows with n vertices we have a large number of possibilities: 4n of them, and the length of each code is 2n. In the case of links, this number grows with the number of components. At the beginning of every
classification the natural problem arises: to classify and enumerate all
possible objects that can be obtained. The same question appeared at the
beginning of the knot theory: to obtain all possible knot shadows. At the
end of the 19th century, the Reverend Thomas Kirkman made tables of alternating knot
diagrams up to 10 crossings (Kirkman, 1885, 1885a). Before
trying to repeat and extend his result (but now with the help of computers),
it will be useful to try to reduce the number of possibilities by minimizing
the number of crossings, and to make the codes more concise.
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