Pairs of knot invariants (Q6629556)

In this paper, the author studies pairs of knot invariants and their relations. For example, it is well known that any Alexander polynomial can be constructed as the Alexander polynomial of an unknotting number one knot; thus, the Alexander polynomial cannot give any information as \(u(K)\geq 2\) where \(u\) is the unknotting number.\N\NIn this paper, the author determines relations between the crossing number and other invariants, for example the unknotting number, the bridge number, the braid index, the genus and the canonical genus.\N\NThe number of knots with crossing number \(n\) is finite. Consequently, the number of knots for the corresponding invariants will also be finite. The author then describes the relation between the crossing number and the above invariants.\N\NThe main theorems are described in the Introduction, and the proofs are given at the end.