1. Petal grid diagrams of torus knots.
- Author
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Lee, Eon-Kyung and Lee, Sang-Jin
- Subjects
- *
TORUS , *KNOT theory , *INTEGERS , *LOGICAL prediction - Abstract
A petal diagram of a knot is a projection with a single multi-crossing such that there are no nested loops. The petal number p (K) of a knot K is the minimum number of loops among all petal diagrams of K. Let T n , s denote the (n , s) -torus knot for relatively prime integers 2 ≤ n < s. Recently, Kim et al. proved that p (T n , s) ≤ 2 s − 2 ⌊ s n ⌋ + 1 whenever s ≡ ± 1 mod n. They conjectured that the inequality holds without the assumption s ≡ ± 1 mod n. They also showed that p (T n , s) = 2 s − 1 whenever 2 ≤ n < s < 2 n and n ≡ 1 mod s − n. Their proofs construct petal grid diagrams for those torus knots. In this paper, we prove the conjecture that p (T n , s) ≤ 2 s − 2 ⌊ s n ⌋ + 1 holds for any 2 ≤ n < s. We also show that p (T n , s) = 2 s − 1 holds for any 2 ≤ n < s < 2 n. Our proofs construct petal grid diagrams for any torus knots. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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