This paper presents a framework to convert argumentative texts into argument knowledge graphs (AKG). The proposed argumentative knowledge representation framework (AKReF) extends the theoretical foundation and enables the AKG to provide a graphical view of the argumentative structure that is easier to understand. Starting with basic annotations of argumentative components (ACs) and argumentative relations (ARs), we enrich the information by constructing a knowledge base (KB) graph with metadata attributes for nodes. Next, we apply modus ponens on premises and inference rules from the KB to form arguments. From these arguments, we create an AKG. The nodes and edges of the AKG have attributes capturing key argumentative features such as the type of premise (e.g., axiom, ordinary premise, assumption), the type of inference rule (e.g., strict, defeasible), preference order over defeasible rules, markers (e.g., "therefore", "however"), and the type of attack (e.g., undercut, rebuttal, undermining). We identify inference rules by locating a specific set of markers, called inference markers (IM). This, in turn, makes it possible to identify undercut attacks previously undetectable in existing datasets. AKG prepares the ground for reasoning tasks, including checking the coherence of arguments and identifying opportunities for revision. For this, it is essential to find indirect relations, many of which are implicit. Our proposed AKG format, with annotated inference rules and modus ponens, helps reasoning models learn the implicit, indirect relations that require inference over arguments and their interconnections. We use an essay from the AAEC dataset to illustrate the framework. We further show its application in complex analyses such as extracting a conflict-free set and a maximal set of admissible arguments.