Knots of genus one or on the number of alternating knots of given genus

Abstract

Abstract. We prove that any non-hyperbolic genus one knot except the trefoil does not have a minimal canonical Seifert surface and that there are only polynomially many in the crossing number positive knots of given genus or given unknotting number.

Genus generators and the positivity of the signature

“…In the proof of Theorem 1.3, Hirasawa's algorithm [16], that lay in the center of the signature Bennequin inequality in [34], finds its application again, this time in combination with the generator theory for diagrams of given canonical genus, initiated in [39], and then developed further in [44,45]. Namely, we use our result of [37] concerning the maximal crossing number of a generating diagram of a given genus.…”

“…This allows to calculate min { σ(K) : K positive, g(K) = g } for any given g by verifying σ on finitely many knots. These knots, the "generators" of [39], can be algorithmically constructed. We will not get into details about this procedure here, since we discussed it extensively elsewhere.…”

“…Theorem 2.9 [39] Reduced (that is, with no nugatory crossings) alternating knot diagrams of given genus decompose into finitely many equivalence classes under flypes and (reversed) applications of antiparallel twists at a crossing:…”

“…For the proof we need to quote one more previous result, about the maximal crossing number c g of generators (see Definition 2.12). In [39], a rather rough estimate on the number d g was given, which was later improved in [44] to d g ≤ 6g − 3. Then in [45] we showed that this inequality is sharp.…”

“…It is easy to see that it suffices also to prove the theorem in case D is prime; the composite case follows easily. For connected sum factors of genus 1 one can use the description in [39]; since all diagrams have a clasp, the genus of D would decrease at most by one.…”