Arc index of pretzel knots of type (-𝑝,𝑞,𝑟)

Abstract

We computed the arc index for some of the pretzel knots K = P ( − p , q , r ) K=P(-p,q,r) with p , q , r ≥ 2 p,q,r\ge 2 , r ≥ q r\geq q and at most one of p , q , r p,q,r is even. If q = 2 q=2 , then the arc index α ( K ) \alpha (K) equals the minimal crossing number c ( K ) c(K) . If p ≥ 3 p\ge 3 and q = 3 q=3 , then α ( K ) = c ( K ) − 1 \alpha (K)=c(K)-1 . If p ≥ 5 p\ge 5 and q = 4 q=4 , then α ( K ) = c ( K ) − 2 \alpha (K)=c(K)-2 .