The scalar class

Every closed fragment's class in the arity-zero Hom space is its parameter value times the class of the empty fragment: this is how the parameter f enters the skein category numerically. The proof is the rank-one argument of Lemma 3.2 read backwards: both rows of the arity-zero pairing are multiples of the empty row, with ratio f(W).

theorem RS.connectionPairing_zero_union {R : ℕ} (f : EdgeRankParameter R) (W G : ClosedFragment) : connectionPairing f.val 0 W G = f.val (W.union G)

The arity-zero pairing is the union value.

theorem RS.ofFragment_eq_smul_empty {R : ℕ} (f : EdgeRankParameter R) (W : ClosedFragment) : HomSpace.ofFragment f.val W = f.val W • HomSpace.ofFragment f.val emptyClosedFragment

The scalar class: the class of a closed fragment in the arity-zero Hom space is its value times the empty class.