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                                                                    EXHIBIT 31.1

                                  CERTIFICATION

I, John J. Puisis, President and Chief Executive Officer of Third Wave
Technologies, Inc. (the "registrant"), certify that:

1.    I have reviewed this Annual Report on Form 10-K of the registrant;

2.    Based on my knowledge, this annual 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
      this report;

3.    Based on my knowledge, the financial statements, and other financial
      information included in this 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 this 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)) and internal
      control over financial reporting as defined in Exchange Act Rules
      13a-15(f) and 15d-15(f) 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 this report is being prepared;

        b.  designed such internal control over financial reporting, or caused
            such internal control over financial reporting to be designed under
            our supervision, to provide reasonable assurance regarding the
            reliability of financial reporting and the preparation of financial
            statements for external purposes in accordance with generally
            accepted accounting principles;

        c.  evaluated the effectiveness of the registrant's disclosure controls
            and procedures and presented in this report our conclusions about
            the effectiveness of the disclosure controls and procedures, as of
            the end of the period covered by this report, based on such
            evaluation; and

        d.  disclosed in this 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: March 15, 2005

                                                        /s/ John J. Puisis
                                                        ------------------------
                                                        John J. Puisis

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