Non-Conjugate Braids Whose Closures Result in the Same Knot

Abstract

For any knot K represented as an n-braid (n ≥ 3), we construct an infinite sequence of pairwise non-conjugate (n + 1)-braids {bm, m ∈ ℕ} representing K.

On non-conjugate braids with the same closure link

“…If the closure link L has a component K , which is the closure of a subbraid β of β of at least three strands, we consider its axis link L(β ). We can move properly the strands of K within β to the right and perform the construction of Shinjo (see proof of Theorem 1.2 of [14]). Then we obtain a family of links that contain infinitely many sublinks, so the family is infinite itself.…”

“…Performing with K the previous construction of [14], we create a sequence of braids {β i }. That is, we set…”

“…choosing the crossings corresponding to the σ ±1 n−1 surrounding the σ n to contain two strands of K (see Lemma 2.1 of [14]). Then β i contain the subbraids γ i corresponding to K, and β i = γ i ∪ γ .…”

“…For reducible braids, Fukunaga [6,7] later extended Morton's original result, showing infinitely many conjugacy classes of 4-braids associated to (2, k)-torus links. Shinjo [14] then obtained a more general theorem for knots.…”

Infinite sequences of mutually non-conjugate surface braids representing same surface-links

“…[2]). This result was generalized in [3,12,13,14,15]. In this paper, we would like to study a similar problem for surface-links and surface braids.…”