The lower bound of the 𝑤-indices of surface links via quandle cocycle invariants

Iwakiri, Masahide

doi:10.1090/S0002-9947-09-04769-2

The lower bound of the $w$-indices of surface links via quandle cocycle invariants
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by Masahide Iwakiri

Trans. Amer. Math. Soc. 362 (2010), 1189-1210

DOI: https://doi.org/10.1090/S0002-9947-09-04769-2

Published electronically: September 23, 2009

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Abstract:

The $w$-index of a surface link $F$ is the minimal number of the triple points of surface braids representing $F$. In this paper, for a given $3$-cocycle, we consider the minimal number of the $w$-indices of surface links whose quandle cocycle invariants associated with $f$ are non-trivial, and denote it $\omega (f)$. In particular, we show that $\omega (\theta _3)=6$ and $\omega (\theta _p)\geq 7$, where $\theta _n$ is Mochizuki’s $3$-cocycle of the dihedral quandle of order $n$ and $p$ is an odd prime integer $\not =3$. As a consequence, for a given non-negative integer $g$, there are surface knots with genus $g$ with the $w$-index $6$.

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