![]() Then the total linking number Lk(D) is obtained by taking half the sum, over all crossings, of contributions from each given by +1 ,1 if the two arcs involved in the crossing belong to di erent components of the link, and 0 if they belong to the same one. It is computed by using a diagram of the link, so we then have to use Reidemeister's theorem to prove that it is independent of this choice of diagram, and consequently really does depend only on the original link. One of the simplest invariants that can actually be computed easily is the linking number of an oriented link. De ne the human number (!) of a knot to be the minimal number of people it takes (holding hands in a chain) to make the knot - what is it for the trefoil and gure-eight? 3.2. Show that the only knots with 4 or 5 arcs are unknots, and show thus that the trefoil has stick number 6. De ne the stick number of a knot to be the minimal number of arc segments with which it can be built. ![]() ![]() The number of components (L) of a link L is an invariant (since wiggling via -moves does not change it, it does depend only on the equivalence class of link). This is an invariant by de nition, but at this stage the only crossing number we can actually compute is that of the unknot, namely zero! Example 3.1.7. The crossing number c(K) is the minimal number of crossings occurring in any diagram of the knot K. Link invariants, oriented link invariants, and so on (for all the di erent types of knotty things we might consider) are de ned and used similarly. As a trivial example, the function i which takes the value 0 on all knots is a valid invariant but which is totally useless! Better examples will be given below. Warning: the de nition does not work in reverse: if two knots have equal invariants then they are not necessarily equivalent. Therefore if i(K) 6= i(K 0 ) then K K 0 cannot be equivalent they have been distinguished by i. The de nition says that if K = K 0 then i(K) =i(K 0 ). The function of an invariant istodistinguish (i.e. The most common invariants are integer-valued, but they might havevalues in the rationals Q, a polynomial ring Z, a Laurent polynomial ring (negative powers of x allowed) Z, or even be functions which assign to any knot a group (thought of up to isomorphism). We have not yet speci ed what kind of values an invariant should take. A knot invariant is any function i of knots which depends only on their equivalence classes. Now that we have Reidemeister's theorem, we can at last construct some invariants and use them to prove that certain knots and links are inequivalent.
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