It's somewhat similar to "but" in meaning. 这我倒不知道 means that while I know that, I don't know this. I guess the context may be: A: .... (A tells/answers B something, showing that he knows about it) B: 那你知道召藩入京勤王这个主意是谁出的吗？(B asks A another question related something A has just told B) A: 这我倒不知道。(A means that while he knows that, he knows this) The word "那" that comes before "你知道" also indicates that A has just told B something he knows.

someone asks you a question, and u know that, but then he asks you another question, u dont know. then you can use:倒 for example, -你知道俄罗斯的首都吗？ -我知道，是莫斯科 -那你知道莫斯科有多少人口吗？ -这我倒不知道。 "倒“ means: oh, i know the first question BUT i dont know this one.

倒？or 到？ It means that I don't know further. Or somebody supposed to know, but actually they don't. Ex: A：你知道小丽要出国了么？ B：知道啊。 A：她什么时候走呢？ B：这我倒（是）不知道。

yeah, just like what comrade brook said, it is a transitional word. You know, 倒means opposite, here it means the truth is opposite the case something should be, or the truth is opposite what or how people think it to be. Some other examples: 本以为他的能力远不如我，想不到，这次倒是我输了。 明明是你先骂的我，你倒还说我不文明！ OK，I'd better refer to 新华字典。 倒 has two tones, a third one and a forth one, here it should be the forth one. I guess you know it. According to the dictionary, there is a meaning of 倒 with the forth tone. Here it comes: 却。example: 东西倒不坏，就是旧了点。（ I think it is a inverted sentence,haha)

If it is supposed that I should know it but I really do not know, then I say 这我倒不知道, which is equal to 这我倒是不知道. In the situation, the word 倒 means but, and, yet, nevertheless and on the contrary.