Correction: Diamantis et al. Topology and Algebra of Bonded Knots and Braids. Mathematics 2025, 13, 3260

Diamantis, Ioannis; Kauffman, Louis H.; Lambropoulou, Sofia

doi:10.3390/math14040610

Text Correction

There was an omission in the original publication [1]. A correction has been made to Section 6.3, in the statement of Theorem 8. The same correction has been made to Section 9.2, in the statement of Theorem 13. Namely, the following relation has been added in the presentations in both stated theorems:

b_{1} {(σ_{2} σ_{1} σ_{3} σ_{2})}^{- 1} b_{1} (σ_{2} σ_{1} σ_{3} σ_{2}) = {(σ_{2} σ_{1} σ_{3} σ_{2})}^{- 1} b_{1} (σ_{2} σ_{1} σ_{3} σ_{2}) b_{1}

Although we explain just above Theorem 8 the method by which one obtains in the statement of Theorem 8 the irredundant presentation of the tight bonded braid monoid

B B_{n}

with only the first elementary bond,

b_{1}

, from its full presentation with all elementary bonds, given in the statement of Theorem 6, yet, the above relation reflecting the far commuting of elementary bonds (

b_{i} b_{j} = b_{j} b_{i} f o r | i - j | > 1

) was omitted in Theorem 8. Then, exactly the same omission followed in the irredundant presentation of the enhanced tight bonded braid group

E B_{n}

, in the statement of Theorem 13. So, the complete statements of Theorems 8 and 13 read as follows:

Theorem 8.

The tight bonded braid monoid

B B_{n}

admits the following irredundant presentation: it is generated by the classical braid generators and their inverses

σ_{1}^{\pm 1}, \dots, σ_{n - 1}^{\pm 1}

and a single bond generator

b_{1}

, subject to the following relations:

\begin{matrix} σ_{i} σ_{j} & = & σ_{j} σ_{i} & for | i - j | > 1, \\ σ_{i} σ_{i + 1} σ_{i} & = & σ_{i + 1} σ_{i} σ_{i + 1} & for all i, \\ b_{1} {σ_{j}}^{\pm 1} & = & {σ_{j}}^{\pm 1} b_{1} & for j > 2, \\ b_{1} {σ_{1}}^{\pm 1} & = & {σ_{1}}^{\pm 1} b_{1} \\ b_{1} (σ_{2} σ_{1}) (σ_{1} σ_{2}) & = & (σ_{2} σ_{1}) (σ_{1} σ_{2}) b_{1} \\ b_{1} {(σ_{2} σ_{1} σ_{3} σ_{2})}^{- 1} b_{1} (σ_{2} σ_{1} σ_{3} σ_{2}) & = & {(σ_{2} σ_{1} σ_{3} σ_{2})}^{- 1} b_{1} (σ_{2} σ_{1} σ_{3} σ_{2}) b_{1} . \end{matrix}

Theorem 13.

The enhanced tight bonded braid group

E B_{n}

admits an irredundant presentation with generators the classical invertible braid generators

σ_{1}, \dots, σ_{n - 1}

together with a single attracting bond generator

b_{1}

and its inverse

b_{1}^{- 1}

, subject to the following relations:

\begin{matrix} σ_{i} σ_{j} & = & σ_{j} σ_{i} & for | i - j | > 1, \\ σ_{i} σ_{i + 1} σ_{i} & = & σ_{i + 1} σ_{i} σ_{i + 1} & for all i, \\ b_{1} σ_{j} & = & σ_{j} b_{1} & for j > 2, \\ b_{1} σ_{1} & = & σ_{1} b_{1} \\ b_{1} (σ_{2} σ_{1}) (σ_{1} σ_{2}) & = & (σ_{2} σ_{1}) (σ_{1} σ_{2}) b_{1} \\ b_{1} {(σ_{2} σ_{1} σ_{3} σ_{2})}^{- 1} b_{1} (σ_{2} σ_{1} σ_{3} σ_{2}) & = & {(σ_{2} σ_{1} σ_{3} σ_{2})}^{- 1} b_{1} (σ_{2} σ_{1} σ_{3} σ_{2}) b_{1} . \end{matrix}

In both statements, of Theorem 8 and of Theorem 13, we also stressed, for full clarity and self-containedness, the invertibility of the classical braid group generators, although this was not an omission. It was already explicitly stated in the full presentation given in Theorem 6 and the classical braid group is a very well-known mathematical object. Namely, in the statement of Theorem 8 we added “and their inverses” as well as the inverse generators of the classical braid group generators, and in the statement of Theorem 13 we added the word “invertible” for the classical braid group generators.

The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.

Reference

Diamantis, I.; Kauffman, L.H.; Lambropoulou, S. Topology and Algebra of Bonded Knots and Braids. Mathematics 2025, 13, 3260. [Google Scholar] [CrossRef] [Scilit]

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