TITLE: Given a knot, what's the minimal genus of a torus the knot is embeddable on?
QUESTION [7 upvotes]: An n-embeddability definition appears towards the end of the section 5.1 Torus knots of the Knot book by C. C. Adams:

A knot $K$ is an $n$-embeddable knot if $K$ can be placed on a genus $n$ standardly embedded surface without crossings, but $K$ cannot be placed on any standardly embedded surface of lower genus without crossings.

Later, an upper bound is imposed on $n$:

A knot with bridge number $b$ is an $n$-embeddable knot where $n \leq b$.

However, so far I failed to find a solution how to embed the $6_3$ knot (bridge number $=2$), on a $2$-torus. The best I can do is a $3$-torus:

Hence my questions, all related to $n$:

Is the bridge number $b$ really an upper bound for $n$?
Is there a knot invariant which would serve as a lower bound for $n$?
Is there a knot invariant matching $n$? (Crosscap gets close, but fails e.g. for the $7_4$ knot.)

Alternatively, I'd be humbled and satisfied by a $2$-torus solution for the $6_3$ knot.
P.S.: If it's of any help, here's the fundamental polygon of the front-counterclockwise cut:

REPLY [3 votes]: To add on to Kyle Miller's answer, the relation between tunnel number and $n$-embeddable invariants is $t(K)\leq n $. There is another invariant called the genus $g$ bridge number, $b_g(k)$, which generalizes bridge knot to a family of invariants, and $b_0(K)$ is the classical bridge number.  (There are a few definitions that vary slightly here depending on the author.)  If $b_g(K) = 0$ for some $g$, then $K$ embeds on a genus $g$ surface and is primitive ( but you can mostly ignore this at the moment.)  From here, you can get a better bounds on tunnel number, $t(K) \leq b_g(K) +g -1$.
Now, for the question you ask about Bridge number being an upper bound for $n$-embeddable, you can see this by the fact that $b_g(K)\geq b_{g+1}(K) +1$.  In other words, genus $g$ bridge numbers must decrease at every step.  So if $b(K) = b_0(K) = n$, then we must have $b_n(K) = 0$, and $K$ embeds on a genus $n$ surface, but maybe it can be embedded on a lower genus link.  
Sometimes $n$-embeddable is referred to as the Heegaard genus of a knot, $h(K)$.  
Last, again to follow Kyle's answer, the original definition of bridge number was the minimum number of "bridges" a diagram can have, which are any arc which has only over crossings. Kyle's fourth picture is an example of this.  Every $n$ bridge knot has a similar diagram which has $n$ arcs that go over the rest of the knot. Then we can make a surface that has a handle for each bridge.