Loop manipulation subgroup of the braid group

Recently, I came across a subgroup of the braid group $$B_{2n}$$ that I'm calling the "loop manipulation" group $$H_n$$. The idea is that we treat pairs of adjacent strands in the braid group as "loops" $$i = 1, \dots, n$$. It is generated by the following elements:

• $$T_i$$ for $$i = 1, \dots, n$$,
• $$C_i$$ for $$i = 1, \dots, n-1$$
• $$P_i$$ for $$i = 1, \dots, n-1$$.

The nomenclature is meant to suggest that $$T_i$$ is a "twist", $$C_i$$ is a "cross", and $$P_i$$ is a "pass". Here is an illustration of the generators: The loop manipulation generators. (Also, there is a copy on my website here.) Generally, $$H_n$$ has a lot in common with the braid group $$B_n$$. The $$C_i$$ generators are analogous to the the usual generators $$\sigma_i$$ of the braid group $$B_{n}$$ so we have:

1. The usual braid relation: $$C_i C_{i+1} C_i = C_{i+1}C_iC_{i+1}$$ for $$i = 1, \dots, n-1$$.
2. The usual braid commutation relation: If $$|i - j| \geq 2$$ then $$C_iC_j = C_jC_i$$

Playing with lots of little diagrams has convinced me of the following relations:

1. $$T_iT_j = T_jT_i$$ for all $$i$$ and $$j$$.
2. $$T_iC_i = C_iT_{i+1}$$ for $$i = 1, \dots, n-1$$.
3. $$T_iP_i = P_iT_{i+1}$$ for $$i = 1, \dots, n-1$$.

I've got two questions about $$H_n$$:

1. Do the relations above determine $$H_n$$ uniquely? To put it formally: If an abstract group generated by $$T_i$$, $$C_i$$, and $$P_i$$ satisfied these relations, would it be isomorphic to $$H_n$$?
2. Does this group appear in the literature anywhere?
• If I've understood things correctly, you need to add the following relations: $P_i C_{i+1} C_i = C_{i+1} C_i P_{i+1}$, and $P_i P_{i+1} C_i = C_{i+1} P_i P_{i+1}$, and $P_i P_{i+1} P_i = P_{i+1} P_i P_{i+1}$. Commented Apr 1 at 17:37
• Thanks. These seem like straightforward generalizations of the usual braid relation. How can I be sure that this list is complete? Commented Apr 1 at 17:59
• You're thinking of $T_i = \sigma_{2i-1}$ every second braid (twist)? Commented Apr 1 at 20:10
• I've seen this referred to as the Hilden subgroup of $B_{2n}$. E.g: arXiv:0706.4421 Theorem 5.3 gives a presentation in terms of your generators and 14 relations. Commented Apr 2 at 1:27
• If you'd invert one of the $C_i$'s in equation (4) (which I understand you don't want to do) then the subgroup generated by the $C_i$ and $T_j$ is an affine braid group. More precisely, replacing your $C_i$ by something that's usually called $T_i$, which in addition obeys the quadratic relation $(T_i - q)(T_i + q^{-1})=0$, and writing $Y_i$ for your $T_i$, instead by gives (the Bernstein–Zelevinsky presentation of) the affine Hecke algebra of type $GL_n$. (I'm not aware of an analogue of your $P_i$ in that case.) Commented Apr 7 at 14:19

Your group $$H_n$$ is (I believe) called the wicket group by Brendle and Hatcher in their paper Configuration spaces of rings and wickets. They provide a presentation in Proposition 3.6. They also provide two references that appear closely related; these are Hilden's 1975 paper and Tawn's 2008 paper.