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In Bevington's The Necessary Shakespeare, the word used instead is "bond." According to onlinedictionary (see link below), a band is "that which serves as the means of union or connection between persons; a tie." Richard is merely reminding John of Gaunt of the oath John has given to Richard as his anointed king, as well as their bond (a.k.a. band) as uncle and nephew (John of Gaunt was Richard's father's brother).
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