This paper proposes four geometric invariants to be used to eliminate joint variables in closure equations for a 6R manipulator instead of the Gaussian elimination used in other studies. The geometric invariants are determined by the structure of any three consecutive joints in space. They have specific geometric meanings such as angle, length, area, volume. For a general 6R manipulator, its four basic closure equations containing only three angular variables can be directly constructed in a few minutes from the geometric invariants using Maple on a general personal computer. These resulting equations have no extraneous roots and are algebraically independent. Because the basic closure equations are obtained from the geometric invariants, they have the most simple forms and provide a very good chance to solve the input-output equation for the inverse kinematics problem. As a result, we use the set of basic closure equations to derive the symbolic 16th degree input-output equation and compute the input-output equation for a special case with 16 different real solutions. All position and orientation coordinates of the end-effector may be symbolic parameters. The definition of the geometric invariants is independent of joint types and can be applied to manipulators with any serial geometry.