Let h(K), (K), g_1(K), t(K) be the h-genus, Heegaard genus, bridge-1 genus, tunnel number of a knot K in the 3-sphere , respectively. It is known that . 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 , the following three families of knots are infinite:
A_{n}={K| t(K)=n<(K)}, B_{n}={K| g_1(K)=n<h(K)}, C_{n}={K| h(K)=n<g_H(K)}.
This is a joint work with Ruifeng Qiu and Chao Wang.
邹燕清，博士毕业于大连理工大学数学科学学院，现为华东师范大学研究员。主要研究方向为三维流形理论研究，已在J. Topol等期刊上发表论文二十余篇，目前主持国家基金面上项目等多项项目。
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