There exist some mechanisms with variable topologies that have interesting applications, for examples, legged walking machines, mechanical push-button stopper locks, and various toys. A variable kinematic joint is a kinematic joint that is capable of topological variation in a mechanism with variable topology. This work aims at the topological representations and characteristic analysis of variable kinematic joints. During the operation process of a mechanism, the topology states of a variable kinematic joint can be expressed symbolically as the joint sequences, graphically the digraphs, and mathematically the matrices. With the applications of graph theory, it proves that the topological characteristics of variable kinematic joints appeared with the abilities of reversibility, continuity, variability of degrees of freedom, joint homomorphism, contractibility, and expansibility. Two examples are provided for illustrating how the proposed concepts can be used to analyze and synthesize the variable joints. The results of this work provide a logical foundation for the systematic structural synthesis regarding the kinematic joints and mechanisms with variable topologies.

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