I have a workbook with multiple sheets and I want to add only numbers
that are less than zero from a particular cell (K12) across sheets.

This approach is not elegant, but it seems to get the right result in Excel 2010.

The column Sheets!B is used as a workspace.

In B1, put
=IF(A1="",0,MIN(0,INDIRECT("'"&A1&"'"&"!K12",TRUE) ))
and copy down as far as the list can ultimatele reach.

Where the answer is to go, put
=SUM(Sheets!B:B)

If a "running total" is required to showing all weekly totals, put this in a Sheets!C1
=SUM(Sheets!B$1:B1)
and copy down. (Another workspace).

If you need each Sheet to contain the total of all the previous sheets, build on Column C by using this:
=INDEX(Sheets!C:C,MATCH(MID(CELL("filename"),FIND( "]",CELL("filename"))+1,255),Sheets!A:A,0)-1)
in any sheet except the first. The MID(...) part privides the tab name of the sheet in which it occurs.

Hope this helps getting started.