Nugatory Crossing And Linking Number

One can get the idea of"nugatory crossing conjecture"from Kirby's paper titled"problems in low-dimensional topology". When doing research on knots, one of the straightest ways is to do some discussions purely from knot diagrams. On the other hand, however, people can also get the most intrinsic properties about knots from such discussions. Nugatory crossings are fundamental in problems about diagrams, and they are hoped to be as simple as possible.The cases for trivial knot and 2-bridge knots have been proved. The cases for composite knots can also be omitted by a known fact. In the present thesis, assuming a knot diagram contains a nugatory crossing, we consider the tangle obtained by removing a neighborhood of the nugatory crossing. Since the crossing change does not affect the knot type of the origin knot diagram, the diagrams before and after the crossing change share the same value on all knot invariants. Observing the whole family of knot invariants, the Vassiliev invariants, the polynomial invariants and a few others can be associated to crossing change directly.In this thesis, the HOMFLY polynomial, which embraces numerous fine properties and includes two other polynomials as its special cases, is preferred as our tool to recover some properties of nugatory crossings. A helpful property we mentioned above is observed by constructing the compute tree of HOMFLY polynomial for the knot diagram. Smoothing the nugatory crossing leaves us a 2-component link. Emphasizing on the property about the tangle mentioned above and two other facts about HOMFLY polynomial claimed in the thesis, we prove our main result, that is, the linking number of the link is zero. This result esteems the nugatory crossing conjecture and makes the study on nugatory crossing easier.