The symmetry of a knot, *s*, is the number of moves to the right minus the
number of moves to the left.
The knot class contains knots with *s*=0 if is
even; otherwise, the optimal symmetry is -1 or 1.
We estimate, for a given class, the symmetry probability density
, asymptotic in *h*.

The move composition of a knot sequence, and hence its symmetry, corresponds
to the coordinates of the analogous random walk.
We begin by rewriting the evolution equations (1) for the persistent
random walk *given* that the fraction steps are along the
*c*-axis,

These give rise to a walk in which the fraction of steps tends toward , while remaining symmetric between and .

Since we are only interested in the and step composition,
we project the 2-dimensional
walk on to the perpendicular to the *c*-axis, say, the *x*-axis, reducing the
problem to a 1-dimensional persistent random walk of steps
of or .
In this simplified walk, a step to the left is
followed with probability by another step to the left and
by a step to the right; a step to the right is similarly
biased toward the left. The resulting evolution equations are written

where is the conditional probability that the walker is
at *x* on the *n*th step, having just taken a step along the positive
*x*-axis, and *v* is conditioned on a step along the negative *x*-axis.

By the central limit theorem, the distribution of the symmetry of the projected random walk

where if the *i*th step is and
if the *i*th step is , approaches a gaussian for large *m*.
Accordingly, we desire the projected walk's mean and variance.

The evolution equations (15) are symmetric about 0 apart from the initial step, which is always . Observing the possible paths taken in the first few steps of the unprojected walk, it is evident that the first moment satisfies

and, accordingly, the mean is

In what follows, we make use of the local correlation function, . It may be observed that

where denotes the average over .
By considering the general average
as successive averages over , , *etc.*, we have

The second moment may be expressed in terms of the local correlation function as

Separating the sum into *i*=*j* and terms, we have

substituting in (20), it follows that

Since the mean is always bounded by [-1,0], we approximate the variance as

Together with (18), the above expression specifies the distribution of the terminal coordinates of the walk and, consequently, the symmetry of the tie.

Sat Nov 7 16:03:57 GMT 1998