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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: October 29, 2004

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