Jones polynomial of a connected sum
= Jones polynomial of a connected sum
{c}
{title2=$V_{K\#J}(t)=V_K(t)V_J(t)$}
For <knots>, matching <bracket smoothing states> across a <connected sum of knots> joins one state <circle> from each side. Their total <circle> count is therefore the sum of the two counts minus one. The reduced <Kauffman bracket> state weights multiply, and <writhe of a link diagram> adds, proving the formula.