generalization of TM: multihead d-dimensional TM

Question

The snippet below is from the Handbook of Theoretical Computer Science, vol. A. My first question is about a possible typo: the text says:

A multi-head d-dimensional machine is a TM with d-dimensional worktape with $k\geq 1$ work-tape heads.

Should it be $d$ instead of $k$ ?

My second question is about the meaning of the $d$ tuple of integers. When the i-th tape goes in the opposite direction as to the root, the ith coordinate increases and otherwise it decreases, right ?

So there are never negative values in the d-tuple, and all are zero iff all the tapes are at the root, am I correct ?

Finally, why does the finite state memory must control whether 2 heads are on the same node and how this can be derived from the d-tuple ?

What does $\Omega$ in the -4th line mean ?

Answer 1

Should it be $d$ instead of $k$ ?

No, number of dimensions and number of tapes are independent.

When the i-th tape goes in the opposite direction as to the root, the ith coordinate increases and otherwise it decreases, right ?

So there are never negative values in the d-tuple, and all are zero iff all the tapes are at the root, am I correct ?

I think you are confusing tree worktape and multidimensional worktape here. They are distinct models, and there is no "root" in multidimensional model, and in tree model the head position isn't described by $d$-tuple.

(also, tapes are not moving, heads do)

Finally, why does the finite state memory must control whether 2 heads are on the same node and how this can be derived from the d-tuple ?

It doesn't control it, it knows it. Formally, when we define transition table, it can depend on whether heads are on the same node or not (and it allows, for example, to avoid ambiguity when heads are trying to write different values to the same place).

What does $\Omega$ in the -4th line mean ?

Asymptotic notation, definitely should be covered in the book somewhere.