4 Plural.

I've been thinking about the main question in the original post on and off for a few days. All of my efforts have been in the direction of finding enough examples to prove a super-linear lower bound, following Misha's suggestion to use hyperbolic volume. This hasn't worked yet - the problem appears to be tricky! In any case, here is the most appealing of the constructions.

Stick braids

Let $x, y, z$ be the usual coordinates on $\mathbb{R}^3$. Let $D$ be the unit disk in the plane $z = 0$ and let $E$ be the unit disk in the plane $z = 1$. Suppose that $\{a_i\}_1^n \subset D$ is a collection of points. Let $b_i$ be the point in $E$ with the same $x$ and $y$ coordinates as $a_i$. Suppose that $\sigma \in \Sigma_n$ is a permutation. Let $B = B(a, \sigma)$ be the collection of line segments where the $i$'th segment has endpoints $a_i$ and $b_{\sigma(i)}$. If the segments are pairwise disjoint then we call $B$ a stick braid.

It follows that the braid closure of $B$ is a link with stick number at most $5n$. As a concrete example, the $(p,q)$-torus knot can be obtained by placing the points $a_i$ at the $p$'th roots of unity, taking $\sigma(i) = i + q$, modulo $p$, and taking a braid closure.

Hyperbolic volume

Now we must use the Euclidean geometry of the braid $B$ to draw conclusions about the hyperbolic volume of the braid closure. Consider the unit disk $D_t$ in the plane $z = t$. As $t$ varies from $0$ to $1$, the points of intersection $D_t \cap B = \{a_i^t\}$ move along straight lines at speeds depending on the slope of the $i$'th strand. Here is a lie: when two points $a_i^t$ and $a_j^t$ come much closer to each other than they are to any of the other points, then there is a definite contribution to hyperbolic volume. Making this precise (ie, actually true) and then finding a braid $B$ that arranges superlinearly many such meetings would give the desired lower bound.

One way to do this would be to take $n$ sufficently large, $\epsilon$ correspondingly small, and take the points $a_i$ to be a generic $\epsilon$--net in $D$. Choose $\sigma$ to be a random permutation. Let $B = B(a, \sigma)$. Take the braid closure and plug everything into SnapPy. I've not tried to do this yet, but it would at least give some data...

Edit

Before writing the above, I had the idea of generating a random stick knot using outer billiards -- namely, let $P$ and $Q$ be concentric spheres and build a knot by taking segments tangent to $Q$ with endpoints on $P$. This has the virtue that when $Q$ has smallish radius, the expected crossing number will be quadratic. But it seems easier to estimate volume using the braid construction, and it is volume that really matters to us.

And then... after all this thinking and writing, I started poking randomly around the web and found O'Rourke's question on our very own MO. O'Rourke gives a very simple model for random stick knots: just bounce around in a sphere. Thurston suggests The Thurstons suggest that the expected volume grows as $n^{3/2}$.

I've been thinking about the main question in the original post on and off for a few days. All of my efforts have been in the direction of finding enough examples to prove a super-linear lower bound, following Misha's suggestion to use hyperbolic volume. This hasn't worked yet - the problem appears to be tricky! In any case, here is the most appealing of the constructions.

Stick braids

Let $x, y, z$ be the usual coordinates on $\mathbb{R}^3$. Let $D$ be the unit disk in the plane $z = 0$ and let $E$ be the unit disk in the plane $z = 1$. Suppose that $\{a_i\}_1^n \subset D$ is a collection of points. Let $b_i$ be the point in $E$ with the same $x$ and $y$ coordinates as $a_i$. Suppose that $\sigma \in \Sigma_n$ is a permutation. Let $B = B(a, \sigma)$ be the collection of line segments where the $i$'th segment has endpoints $a_i$ and $b_{\sigma(i)}$. If the segments are pairwise disjoint then we call $B$ a stick braid.

It follows that the braid closure of $B$ is a link with stick number at most $5n$. As a concrete example, the $(p,q)$-torus knot can be obtained by placing the points $a_i$ at the $p$'th roots of unity, taking $\sigma(i) = i + q$, modulo $p$, and taking a braid closure.

Hyperbolic volume

Now we must use the Euclidean geometry of the braid $B$ to draw conclusions about the hyperbolic volume of the braid closure. Consider the unit disk $D_t$ in the plane $z = t$. As $t$ varies from $0$ to $1$, the points of intersection $D_t \cap B = \{a_i^t\}$ move along straight lines at speeds depending on the slope of the $i$'th strand. Here is a lie: when two points $a_i^t$ and $a_j^t$ come much closer to each other than they are to any of the other points, then there is a definite contribution to hyperbolic volume. Making this precise (ie, actually true) and then finding a braid $B$ that arranges superlinearly many such meetings would give the desired lower bound.

One way to do this would be to take $n$ sufficently large, $\epsilon$ correspondingly small, and take the points $a_i$ to be a generic $\epsilon$--net in $D$. Choose $\sigma$ to be a random permutation. Let $B = B(a, \sigma)$. Take the braid closure and plug everything into SnapPy. I've not tried to do this yet, but it would at least give some data...

Edit

Before writing the above, I had the idea of generating a random stick knot using outer billiards -- namely, let $P$ and $Q$ be concentric spheres and build a knot by taking segments tangent to $Q$ with endpoints on $P$. This has the virtue that when $Q$ has smallish radius, the expected crossing number will be quadratic. But it seems easier to estimate volume using the braid construction, and it is volume that really matters to us.

And then... after all this thinking, I started poking randomly around the web and found O'Rourke's question on our very own MO. O'Rourke gives a very simple model for random stick knots: just bounce around in a sphere. Thurston says that SnapPea says suggests that the expected volume grows as $n^{3/2}$.

2 Added reference to closely related post.

I've been thinking about the main question in the original post on and off for a few days. All of my efforts have been in the direction of finding enough examples to prove a super-linear lower bound, following Misha's suggestion to use hyperbolic volume. This hasn't worked yet - the problem appears to be tricky! In any case, here is the most appealing of the constructions.

Stick braids

Let $x, y, z$ be the usual coordinates on $\mathbb{R}^3$. Let $D$ be the unit disk in the plane $z = 0$ and let $E$ be the unit disk in the plane $z = 1$. Suppose that $\{a_i\}_1^n \subset D$ is a collection of points. Let $b_i$ be the point in $E$ with the same $x$ and $y$ coordinates as $a_i$. Suppose that $\sigma \in \Sigma_n$ is a permutation. Let $B = B(a, \sigma)$ be the collection of line segments where the $i$'th segment has endpoints $a_i$ and $b_{\sigma(i)}$. If the segments are pairwise disjoint then we call $B$ a stick braid.

It follows that the braid closure of $B$ is a link with stick number at most $5n$. As a concrete example, the $(p,q)$-torus knot can be obtained by placing the points $a_i$ at the $p$'th roots of unity, taking $\sigma(i) = i + q$, modulo $p$, and taking a braid closure.

Hyperbolic volume

Now we must use the Euclidean geometry of the braid $B$ to draw conclusions about the hyperbolic volume of the braid closure. Consider the unit disk $D_t$ in the plane $z = t$. As $t$ varies from $0$ to $1$, the points of intersection $D_t \cap B = \{a_i^t\}$ move along straight lines at speeds depending on the slope of the $i$'th strand. Here is a lie: when two points $a_i^t$ and $a_j^t$ come much closer to each other than they are to any of the other points, then there is a definite contribution to hyperbolic volume. Making this precise (ie, actually true) and then finding a braid $B$ that arranges superlinearly many such meetings would give the desired lower bound.

One way to do this would be to take $n$ sufficently large, $\epsilon$ correspondingly small, and take the points $a_i$ to be a generic $\epsilon$--net in $D$. Choose $\sigma$ to be a random permutation. Let $B = B(a, \sigma)$. Take the braid closure and plug everything into SnapPy. I've not tried to do this yet, but it would at least give some data...

Edit

Before writing the above, I had the idea of generating a random stick knot using outer billiards -- namely, let $P$ and $Q$ be concentric spheres and build a knot by taking segments tangent to $Q$ with endpoints on $P$. This has the virtue that when $Q$ has smallish radius, the expected crossing number will be quadratic. But it seems easier to estimate volume using the braid construction, and it is volume that really matters to us.

And then... after all this thinking, I started poking randomly around the web and found O'Rourke's question on our very own MO. O'Rourke gives a very simple model for random stick knots: just bounce around in a sphere. Thurston says that SnapPea says that the expected volume grows as $n^{3/2}$.

1