= Jones unknot detection problem
{c}
{title2=$V_K(t)=1\ \Longrightarrow\ K\cong U\ ?$}
This asks whether the <Jones polynomial> equal to one characterizes the <unknot> among <knots>. A zero <breadth of a Laurent polynomial> makes $V_K$ a <monomial>; the identities $V_K(1)=1$ and $V'_K(1)=0$ then force that monomial to be one. Thus using Jones breadth as a strictly positive complexity for nontrivial <knots> requires this detection property. To prove the <derivative> identity, differentiate the <Jones polynomial skein relation> at one for a crossing change in a <knot>. Both crossed diagrams have value one, while the smoothed two-component <link> has value minus two by <Jones polynomial evaluation at one>, so $V'_{K_+}(1)=V'_{K_-}(1)$. Crossing changes reduce to the <unknot>, whose <derivative> is zero. The corresponding <Alexander polynomial> detection property is false, as nontrivial untwisted <Whitehead doubles> demonstrate. The Jones question is discussed in https://arxiv.org/abs/2402.02553[research on closed four-strand braids].
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