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                                                                    EXHIBIT 31.1

                                  CERTIFICATION

I, John Puisis, CEO of Third Wave Technologies, Inc. (the "registrant"), certify
that:

1. I have reviewed this Quarterly Report on Form 10-Q (the "Report") of the
   registrant;

2. Based on my knowledge, the Report does not contain any untrue statement of a
   material fact or omit to state a material fact necessary to make the
   statements made, in light of the circumstances under which such statements
   were made, not misleading with respect to the period covered by the Report;

3. Based on my knowledge, the financial statements, and other financial
   information included in the Report, fairly present in all material respects
   the financial condition, results of operations and cash flows of the
   registrant as of, and for, the periods presented in the Report;

4. The registrant's other certifying officer and I are responsible for
   establishing and maintaining disclosure controls and procedures (as defined
   in Exchange Act Rules 13a-15(e) and 15d-15(e)) for the registrant and have:

    (a) designed such disclosure controls and procedures, or caused such
        disclosure controls and procedures to be designed under our supervision,
        to ensure that material information relating to the registrant,
        including its consolidated subsidiaries, is made known to us by others
        within those entities, particularly during the period in which the
        Report is being prepared;

    (b) evaluated the effectiveness of the registrant's disclosure controls and
        procedures and presented in the Report our conclusions about the
        effectiveness of the disclosure controls and procedures, as of the end
        of the period covered by the Report, based on such evaluation; and

    (c) disclosed in the Report any change in the registrant's internal control
        over financial reporting that occurred during the registrant's most
        recent fiscal quarter that has materially affected, or is reasonably
        likely to materially affect, the registrant's internal control over
        financial reporting; and,

5. The registrant's other certifying officer and I have disclosed, based on our
   most recent evaluation of internal control over financial reporting, to the
   registrant's auditors and the audit committee of registrant's board of
   directors (or persons performing the equivalent functions):

    (a) all significant deficiencies and material weaknesses in the design or
        operation of internal control over financial reporting which are
        reasonably likely to adversely affect the registrant's ability to
        record, process, summarize and report financial data; and

    (b) any fraud, whether or not material, that involves management or other
        employees who have a significant role in the registrant's internal
        control over financial reporting.

             Date: August 9, 2004

                                                     /s/ John Puisis
                                                     ---------------------------
                                                     John Puisis, CEO