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Oddly enough I was just thinking about this question a few days ago and nearly posted it to MO. From an old Martin Gardner paper (which I'd vaguely remembered from Malia telling me about it) I found a reference to: MR0133125 J. Shepperd "Braids which can be plaited with their threads tied together at each end." That paper answers the unframed version of your question for arbitrary B_n. The framed case is more complicated and not addressed in that paper. (Jim Tanton gave a Mathcamp class about some open questions related to the framed 4-strand case last summer, unfortunately I didn't attend that class.)

My motivation was some research I'm doing on relatives of $U_q(\mathfrak{g}_2)$, where I needed to understand all 1-dimensional representations of B_3/N_3B_3/T_3. This was easy enough to work out directly, but it made me realize I'd really like to know about all small representations of B_n/N_n B_n/T_n for small n. Essentially this is because "low weight spaces" in ribbon planar algebras (i.e. ribbon categories with a fixed favorite object) always have the structure of a representation of B_n/N_n B_n/T_n (where you're looking at low-weight vectors in the n-box space, i.e. invariants of the nth tensor power of your fixed object).
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Oddly enough I was just thinking about this question a few days ago and nearly posted it to MO. From an old Martin Gardner paper (which I'd vaguely remembered from Malia telling me about it) I found a reference to: MR0133125 J. Shepperd "Braids which can be plaited with their threads tied together at each end." That paper answers the unframed version of your question for arbitrary B_n. The framed case is more complicated and not addressed in that paper. (Jim Tanton gave a Mathcamp class about some open questions related to the framed 4-strand case last summer, unfortunately I didn't attend that class.)

My motivation was some research I'm doing on relatives of $U_q(\mathfrak{g}_2)$, where I needed to understand all 1-dimensional representations of B_3/N_3. This was easy enough to work out directly, but it made me realize I'd really like to know about all small representations of B_n/N_n for small n. Essentially this is because "low weight spaces" in ribbon planar algebras (i.e. ribbon categories with a fixed favorite object) always have the structure of a representation of B_n/N_n (where you're looking at low-weight vectors in the n-box space, i.e. invariants of the nth tensor power of your fixed object).