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Symmetry

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 tex2html_wrap_inline898 contains knots with s=0 if tex2html_wrap_inline1106 is even; otherwise, the optimal symmetry is -1 or 1. We estimate, for a given class, the symmetry probability density tex2html_wrap_inline1138 , 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 tex2html_wrap_inline802 steps are along the c-axis,

  eqnarray213

These give rise to a walk in which the fraction of tex2html_wrap_inline994 steps tends toward tex2html_wrap_inline802 , while remaining symmetric between tex2html_wrap_inline988 and tex2html_wrap_inline948 .

Since we are only interested in the tex2html_wrap_inline988 and tex2html_wrap_inline948 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 tex2html_wrap_inline1162 steps of tex2html_wrap_inline988 or tex2html_wrap_inline948 . In this simplified walk, a step to the left is followed with probability tex2html_wrap_inline1168 by another step to the left and tex2html_wrap_inline1170 by a step to the right; a step to the right is similarly biased toward the left. The resulting evolution equations are written

  eqnarray231

where tex2html_wrap_inline1172 is the conditional probability that the walker is at x on the nth 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

  equation244

where tex2html_wrap_inline1184 if the ith step is tex2html_wrap_inline988 and tex2html_wrap_inline1190 if the ith step is tex2html_wrap_inline948 , 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 tex2html_wrap_inline948 . Observing the possible paths taken in the first few steps of the unprojected walk, it is evident that the first moment tex2html_wrap_inline1200 satisfies

  equation249

and, accordingly, the mean tex2html_wrap_inline1202 is

  equation256

In what follows, we make use of the local correlation function, tex2html_wrap_inline1204 . It may be observed that

  equation261

where tex2html_wrap_inline1206 denotes the average over tex2html_wrap_inline1208 . By considering the general average tex2html_wrap_inline1204 as successive averages over tex2html_wrap_inline1212 , tex2html_wrap_inline1214 , etc., we have

  equation276

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

  equation283

Separating the sum into i=j and tex2html_wrap_inline1218 terms, we have

  equation287

substituting in (20), it follows that

  equation297

Since the mean tex2html_wrap_inline1202 is always bounded by [-1,0], we approximate the variance as

  equation302

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


next up previous
Next: Balance Up: Tie Knots and Random Previous: Shape of Knots

Yong Mao
Sat Nov 7 16:03:57 GMT 1998