In this paper, we investigate three geometrical invariants of knots, the height, the trunk and the representativity. First, we give a counterexample for the conjecture which states that the height is additive under connected sum of knots. We also define the minimal height of a knot and give a potential example which has a gap between the height and the minimal height. Next, we show that the representativity is bounded above by a half of the trunk. We also define the trunk of a tangle and show that if a knot has an essential tangle decomposition, then the representativity is bounded above by half of the trunk of either of the two tangles. Finally, we remark on the difference among Gabai's thin position, ordered thin position and minimal critical position. We also give an example of a knot which bounds an essential non-orientable spanning surface, but has arbitrarily large representativity.
The first author was partially supported by NSF grant DMS 1821254. The second author was partially supported by Grant-in-Aid for Scientific Research (C) (No. 26400097 and 17K05262), Ministry of Education, Culture, Sports, Science and Technology, Japan.
"Height, trunk and representativity of knots." J. Math. Soc. Japan 71 (4) 1105 - 1121, October, 2019. https://doi.org/10.2969/jmsj/80438043