Comparing h-genera, Bridge-1 genera and Heegaard genera of knots

2025/04/27 by Qiu, Ruifeng, Wang, Chao, Zou, Yanqing
#57K10 #57K31 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2504.19118

Abstract

Let h(K), g_H(K), g₁(K), t(K) be the h-genus, Heegaard genus, bridge-1 genus, tunnel number of a knot K in the 3-sphere S³, respectively. It is known that g_H(K)-1=t(K)≤ g₁(K)≤ h(K)≤ g_H(K). A natural question arises: when do these invariants become equal? We provide the necessary and sufficient conditions for equality and use these to show that for each integer n≥ 1, the following three families of knots are infinite: \begineqnarray A_n=\K| t(K)=n

Citations

• On tunnel numbers of a cable knot and its companion
• 3-manifolding admitting locally large distance 2 Heegaard splittings
• On the degeneration of tunnel numbers under connected sum
• The geometry of the disk complex
• Heegaard splittings with large subsurface distances
• High distance knots
• Alternate Heegaard genus bounds distance
• Geometry of the complex of curves II: Hierarchical structure
• 3-manifolds as viewed from the curve complex
• 3-Manifolds as viewed from the curve complex

Related

• On tunnel number one knots that are not (1,n)
• On Heegaard splittings of glued 3-manifolds
• On Heegaard splittings of knot exteriors with tunnel number degenerations
• Genus bounds bridge number for high distance knots
• Genus two Heegaard splittings of exteriors of 1-genus 1-bridge knots II