Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory. Vassily Olegovich Manturov

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n be an integer. Consider the free product G of n copies of the group with generators a₁, . . . , an. Denote by Ei the right residue classes in G by {ai}; i.e., g₁, g₂ ∈ G represent the same element of Ei if and only if for some k.

      Definition 2.13. An n-system is a set of elements e₁ ∈ E₁, . . . , en, ∈ En;

      An ordered n-system is an n-system together with a permutation from Sn.

      Proposition 2.1. There exists an injective map from equivalence classes of admissible systems of curves to ordered n-systems.

      Proof. Since the permutation for equivalent admissible systems of curves is the same, we can fix the permutation s ∈ Sn and consider only equivalence classes of admissible systems of curves with permutation s (i.e., with all lower points fixed depending on the upper points in accordance with s). Thus we only have to show that there exists an injective map from the set of admissible systems of n curves with fixed lower points to n-systems.

      To complete the proof of the proposition, it suffices to prove the following.

      Lemma 2.4. Equivalence classes of curves with fixed points (i, 1) and (j, 0) are in one-to-one correspondence with Ei.

      Proof. Denote by Pn. Obviously, π₁(Pn) ≅ G. Consider a small circle C centred at (i, 1) for some i with the lowest point X on it. Let ρ be a curve with endpoints (i, 1) and (j, 0). Without loss of generality, assume that ρ intersects C in a finite number of points. Let Q be the first such point that one meets while walking along ρ from (i, 1) to (j, 0). Thus we obtain a curve ρ′ coming from C to (j, 0). Now, let us construct an element of π₁(Pn, X). First it comes from X to Q along C clockwise. Then it goes along ρ until (j, 0). After this, it goes along Ox to the point (i, 0). Then it goes vertically upwards till the intersection with C in X. Denote the constructed element by W(ρ).

      If we deform ρ outside C, then we obtain a continuous deformation of the loop, thus W(ρ) stays the same as an element of the fundamental group. The deformations of ρ inside C might change W(ρ) by multiplying it by ai on the left side. So, we have constructed a map from equivalence classes of curves with fixed points (i, 1) and (j, 0) to Ei.

      The inverse map can be easily constructed as follows. Let W be an element of π₁(Pn, X). Consider a loop L representing W. Now consider the curve L′ that first goes from (i, 1) to X vertically, then goes along L′, after this goes vertically downwards until (i, 0) and finally, horizontally until (j, 0). Obviously, W(L′) = W. It is easy to see that for different representatives L of W we obtain the same L′. Besides, for L₁ = aiL₂, the curves and are isotopie This completes the proof of the lemma.

      Thus, for a fixed permutation s, admissible systems of curves can be uniquely encoded by n-systems, which completes the proof of the proposition.

      Now, we see that this invariant is a quite simple object: elements of Ei can be easily compared.

      Let us describe the algebraic construction of the invariant f in more detail.

      Let β be a braid word, written as a product of generators , where each εj is either +1 or −1; 1 ≤ ij ≤ n − 1 and σ₁, . . . , σ_n−1 are the standard generators of the braid group Br(n).

      We are going to construct the n-system step-by-step while writing the word β. First, let us write n empty words (in the alphabet a₁, . . . , an). Let the first letter of β be σj. Then all words except for the word e_j+1 should stay the same (i.e., empty), and the word e_j+1 becomes . If the first crossing is negative; i.e., then all words except ej stay the same and ej converts to a_j+1. While considering each next crossing, we do the following. Let the crossing be . Let p and q be the numbers of strands coming from the left side and from the right side respectively. If this crossing is positive; i.e., σj, then all words except eq stay the same, and eq becomes . If it is negative, then all crossings except ep stay the same, and ep becomes . After processing all the crossings, we get the desired n-system.

      Example 2.1. For the trivial braid written as the construction operation works as follows:

      

      A priori these words may be non-trivial; they must only represent trivial residue classes, say, .

      However, it is not the case.

      Proposition 2.2. For the trivial braid, the algebraic algorithm described above gives trivial words.

      Proof. Indeed, the algebraic number of occurrences of ai in the word ei equals zero. This can be easily proved by induction on the number of crossings. In the initial position all words are trivial. The induction step is obvious. Thus, the final word ei equals , where p = 0.

      From this approach, one can easily obtain the well known invariant (action) as follows. Instead of a set of n words e₁, . . . , en, one can consider the words .

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