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In earlier work the Kauffman bracket polynomial was extended to an invariantof marked graphs, i.e., looped graphs whose vertices have been partitioned intotwo classes (marked and not marked). The marked-graph bracket polynomial isreadily modified to handle graphs with weighted vertices. We present formulasthat simplify the computation of this weighted bracket for graphs that containtwin vertices or are constructed using graph composition, and we show thatgraph composition corresponds to the construction of a link diagram fromtangles.

A bracket polynomial for graphs. III. Vertex weights