HMM-based speech recognition/synthesis: state representation of monophones and triphones
My current understanding is that each of the phones in a triphone corresponds to each of the states. 


That's not correct. 

Each phone, regardless of the context dependency, has three states: "start", "in-between", and "end". There are always transitions from the previous state to the next one and to itself. This defines the structure of HMM. Now context-dependency means that for each triphone we know its left and right context (context-independent phone). Let's take a look at what phone AX could look like in both CI and CD HMMs for the word "one [W AX N]".

When context-independent phones are used AX is begin split it into three states - _AX_B_, _AX_I, _AX_E - to construct the corresponding HMM:



In case of context-dependent phones AX actually doesn't exist any more. Instead we have many new phones that store information of phones to the left of it and to the right of it. Thus in the given word the neighbours of AX are W(left) and N(right), so HMM for phone W-AX-N is constructed: