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The purpose of this erratum is to fill a gap in the proof of the `CompositeBraid Theorem' in the manuscript "Studying Links Via Closed Braids IV:Composite Links and Split Links (SLVCB-IV)", Inventiones Math, \{bf 102\} Fasc.1 (1990), 115-139. The statement of the theorem is unchanged. The gap occurs onpage 135, lines $13^-$ to $11^-$, where we fail to consider the case: $V_2 = 4,V_4 > 0, V_j=0$ if $j not= 2,4,$ and all 4 vertices of valence 2 are bad.  At the end of this Erratum we make some brief remarks on the literature, asit has evolved during the 14 years between the publication of SLVCB-IV and thesubmission of this Erratum.Comment: 6 pages, 4 figures. This is an Erratum to "Studying Links Vai Closed  Braids IV: Composite Links and Split Links", Inventiones Math., 102 Facs. 1  (190), 115-13

Studying links via closed braids IV: composite links and split links