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Edit: In response to Ilya N's comment below, I've made this into its own question.

A Vassiliev (or finite type) invariant of type $\leq n$ is any knot invariant that vanishes on singular knots with $> n$ self-intersections. The space of all Vassiliev invariants is a filtered bialgebra $V$ (filtered by type). The corresponding associated-graded bialgebra $W$ (of "weight systems") has been well-studied (some names: Kontsevich, Bar-Natan, Vaintrob, and I'm sure there are others I haven't read yet) and in fact is more-or-less completely understood (e.g. Hinich and Vaintrob, 2002, "Cyclic operads and algebra of chord diagrams", MR1913297, where its graded dual $A$ of "chord diagrams" is described as a sort of universal enveloping algebra). In fact, this algebra $W$ is Hopf. I learned from this question that this implies that the bialgebra $V$ of Vassiliev invariants is also Hopf. Thus it is a Hopf sub-bialgebra of the algebra $I$ of knot invariants.

Edit: In response to Ilya N's comment below, I've made this into its own question.

A Vassiliev (or finite type) invariant of type $\leq n$ is any knot invariant that vanishes on singular knots with $> n$ self-intersections. The space of all Vassiliev invariants is a filtered bialgebra $V$ (filtered by type). The corresponding associated-graded bialgebra $W$ (of "weight systems") has been well-studied (some names: Kontsevich, Bar-Natan, Vaintrob, and I'm sure there are others I haven't read yet) and in fact is more-or-less completely understood (e.g. Hinich and Vaintrob, 2002, "Cyclic operads and algebra of chord diagrams", MR1913297, where its graded dual $A$ of "chord diagrams" is described as a sort of universal enveloping algebra). In fact, this algebra $W$ is Hopf. I learned from this question that this implies that the bialgebra $V$ of Vassiliev invariants is also Hopf. Thus it is a Hopf sub-bialgebra of the algebra $I$ of knot invariants.

Recall that a (oriented) knot is a smoothly embedded circle $S^1$ in $\mathbb R^3$, up to some natural equivalence relation (which is not quite trivial to write down). The collection of oriented knots has a binary operation called connected sum: if $K_1,K_2$ are knots, then $K_1 \\# K_2$ is formed by spatially separating the knots, then connecting them by a very thin rectangle, which is glued on so that all the orientations are correct. Connect sum is commutative and associative, making the space of knots into a commutative monoid. In fact, by a theorem of Schubert, this is the free commutative monoid on countably many generators. A ($\mathbb C$-valued) knot invariant is a $\mathbb C$-valued function on this monoid; under "pointwise" multiplication, the space of knot invariants is a commutative algebra $I$, and $\\#$ makes $I$ into a cocommutative bialgebra. I.e. $I$ is a commutative monoid object in $(\text{CAlg})^{\rm{op}}$, where $\text{CAlg}$ is the category of commutative algebras.

A Vassiliev (or finite type) invariant of type $\leq n$ is any knot invariant that vanishes on singular knots with $> n$ self-intersections. The space of all Vassiliev invariants is a filtered bialgebra $V$ (filtered by type). The corresponding associated-graded bialgebra $W$ (of "weight systems") has been well-studied (some names: Kontsevich, Bar-Natan, Vaintrob, and I'm sure there are others I haven't read yet) and in fact is more-or-less completely understood (e.g. Hinich and Vaintrob, 2002, "Cyclic operads and algebra of chord diagrams", MR1913297, where its graded dual $A$ of "chord diagrams" is described as a sort of universal enveloping algebra). In fact, this algebra $W$ is Hopf. I learned from this question that this implies that the bialgebra $V$ of Vassiliev invariants is also Hopf. Thus it is a Hopf sub-bialgebra of the algebra $I$ of knot invariants.

Recall that a (oriented) knot is a smoothly embedded circle $S^1$ in $\mathbb R^3$, up to some natural equivalence relation (which is not quite trivial to write down). The collection of oriented knots has a binary operation called connected sum: if $K_1,K_2$ are knots, then $K_1 \# K_2$ is formed by spatially separating the knots, then connecting them by a very thin rectangle, which is glued on so that all the orientations are correct. Connect sum is commutative and associative, making the space of knots into a commutative monoid. In fact, by a theorem of Schubert, this is the free commutative monoid on countably many generators. A ($\mathbb C$-valued) knot invariant is a $\mathbb C$-valued function on this monoid; under "pointwise" multiplication, the space of knot invariants is a commutative algebra $I$, and $\#$ makes $I$ into a cocommutative bialgebra. I.e. $I$ is a commutative monoid object in $(\text{CAlg})^{\rm{op}}$, where $\text{CAlg}$ is the category of commutative algebras.

A Vassiliev (or finite type) invariant of type $\leq n$ is any knot invariant that vanishes on singular knots with $> n$ self-intersections. The space of all Vassiliev invariants is a filtered bialgebra $V$ (filtered by type). The corresponding associated-graded bialgebra $W$ (of "weight systems") has been well-studied (some names: Kontsevich, Bar-Natan, Vaintrob, and I'm sure there are others I haven't read yet) and in fact is more-or-less completely understood (e.g. Hinich and Vaintrob, 2002, "Cyclic operads and algebra of chord diagrams", MR1913297, where its graded dual $A$ of "chord diagrams" is described as a sort of universal enveloping algebra). In fact, this algebra $W$ is Hopf. I learned from this question that this implies that the bialgebra $V$ of Vassiliev invariants is also Hopf. Thus it is a Hopf sub-bialgebra of the algebra $I$ of knot invariants.

If Vassiliev invariants do separate knots, are there still more global points of $V$ than just the free abelian group on countably many generators (i.e. the group generated by the free monoid of knots)? Yes: the singular knots. What else is there?

If Vassiliev invariants do separate knots, are there still more global points of $V$ than just the free abelian group on countably many generators (i.e. the group generated by the free monoid of knots)? Yes: the singular knots. (Edit: The rule for being a global point is that you can evaluate any knot invariant at it, and that the value of the invariant given by pointwise multiplication on knots is the multiplication of the values at the global point. Let $K_0$ be a singular knot with one crossing and with non-singular blow-ups $K_+$ and $K_-$, and let $f,g$ be two knot invariants. Then $$\begin{aligned} (f\cdot g)(K_0) & = (f\cdot g)(K_+) - (f\cdot g)(K_-) = f(K_+)\cdot g(K_+) - f(K_-)\cdot g(K_-) \neq \\\\ f(K_0) \cdot g(K_0) & = f(K_+)\cdot g(K_+) - f(K_+)\cdot g(K_-) - f(K_-)\cdot g(K_+) + f(K_-)\cdot g(K_-)\end{aligned}$$.) What else is there?