Symmetric Representations Of Knot Groups

A knot is a closed curve in the three-dimensional space,which is homeomorphism image of a circle in three-dimensional space.The central question of knot theory:Given a knot or link,how to determine whether it is unknot,that is,whether the knot is equivalent to the trivial knot or link.For any two given knots or links,how to determine whether they are the same.This thesis studies knot from a group theoretic point of view.A new method has been presented to construct symmetric representations of knot groups.On the other hand,the author also gives a computer algorithm,and uses it to give a full list of symmetric representations of some special knots to certain symmetric groups.Conjugation induces an equivalence relation among representations.In the last part,a full list of the above symmetric representations up to this equivalence relation is given.