difference between two electrodes in the same position of different TMs: these
are basically not resolved, mainly due to the degeneracy of their
$\textbf{A}(q)$ coefficients.

Figure 13: Examples of distributions of $S_{\alpha_{\mathrm{UC}}}$ (orange)
and $S_{\alpha_{\mathrm{C}}}$ (black) obtained from the MCMC fit. At both
frequencies the uncorrelated electrode gain noise $S_{\alpha_{\mathrm{UC}}}$
is robustly resolved from the experimental data and largely independent to any
prior assumptions on the distribution, while the lower tail of the posterior
for “board correlated” gain noise $S_{\alpha_{\mathrm{C}}}$ is only limited,
in the lower values, by the prior. To allow better visualization, the former
is normalized to 1, the latter to 4. The dashed line instead gives an idea of
the $S_{\alpha_{C}}$ cutoff: its values have no particular meaning in this
plot and should only help visualizing its position and shape. At right are
distributions, at the same two frequencies, for the partitioning of the levels
of uncorrelated gain noise between different groups of electrodes.

In this paper, estimates of the parameters are often reported by points with
error bars (the central point is the 50 percentile while bars indicate the
equally-tailed 68% confidence interval), that can be seen for example in Figs.
5 and 6. While we consider our priors to be both conservative and physically
motivated, we give a visual indication of the data points for model parameters
in the cases where the lower limit of the posterior distribution is strongly
dependent on the prior assumptions and not strongly constrained by the
experimental data. We can define an effective lower limit
$S_{ll}\left(\bar{S}_{co}\right)$ as the PSD value at the $-1\sigma$ level
(15.9% percentile) of the distribution. In presence of important tails like
$S_{\alpha_{\mathrm{C}}}$ in Figure 13, if the prior cutoff ($\bar{S}_{co}$)
is lowered of a factor 10, the lower limit $S_{ll}\left(\bar{S}_{co}\right)$
behaves almost exactly as if the posterior distribution were uniform (in log),
decreasing by $\sim 86$% in linear scale. We adopted, as a rule of thumb, the
criteria for marking with a dash-dot line, to indicate sensitivity to the
lower-cutoff prior, all the parameters estimates for which

$\frac{S_{ll}(\bar{S}_{co}/10)}{S_{ll}(\bar{S}_{co})}<\frac{1}{2}\,.$ (67)

For these parameters, the distribution upper limit is instead much less
sensitive to the prior cutoff choice, varying typically by order several
percent and never more than 10% for a factor 10 increase in the upper-cutoff.

The regularity of the results obtained analysing each frequency independently
suggest that the whole behaviour can be safely described also using smooth
analytical functions of the frequency $f$, specifically

$\displaystyle S_{\alpha_{i}}(f)$
$\displaystyle=\frac{\epsilon_{i}}{f^{2}+f_{\mathrm{cut}}^{2}}+\frac{\rho_{i}}{f}$
$\displaystyle S_{\alpha_{ij}}(f)$
$\displaystyle=\frac{\epsilon_{ij}}{f^{2}+f_{\mathrm{cut}}^{2}}+\frac{\rho_{ij}}{f}$
$\displaystyle S_{\Delta g}^{bg}(f)$ $\displaystyle=\frac{\epsilon_{\Delta
g}^{bg}}{f^{2}}+\rho_{\Delta g}^{bg}$ $\displaystyle
S_{\Delta\gamma_{\phi}}^{bg}(f)$
$\displaystyle=\frac{\epsilon_{\Delta\gamma_{\phi}}^{bg}}{f}+\rho_{\Delta\gamma_{\phi}}^{bg}f^{2}\,.$
(68)

The analytic form was chosen empirically based on the measured PSD. After re-
parametrizing $\epsilon,\,\rho$ in Eqn. 68 similarly to Eqn. 64, the MCMC runs
on a total of 26 parameters. While the previous fit was performed at a single
frequency, here all bins are fit at once, namely
$\hat{\textbf{D}}=\\{\hat{\textbf{S}}(f,q)\\}_{f,q}$ at the frequencies,
roughly 50, set by the standard Welch periodogram length. The likelihood
function is exactly the same as Eqn. 63, but multiplied over all frequencies,
since they are all independent:

$L(\hat{\textbf{D}}|\textbf{M})=\prod_{f,q}p_{f,q}(\hat{\textbf{S}}|\textbf{M})$
(69)

Also in this case, logarithmic uniform priors were put on $\epsilon,\rho$
parameters related to average uncorrelated noise, correlated noise and
background, while uniform priors were used for $\mu$’s expressing difference
between different noise generators, and $\xi$ (cross-background parameter). A
lower cutoff was needed just for the two parameters associated with correlated
noise, and was fixed to 1/100 of data-sheet values.

To test the goodness of our fits, we employed a posterior predictive check
[36]. Assuming the distribution of the model parameters (M) obtained from the
MCMC, we derive the distributions for $\hat{\textbf{D}}$ and then evaluate the
compatibility with the the experimental data. To do this, for each value of M
accepted in the Markov Chain we sample values of $\textbf{X}_{s}$ – the
Fourier components in $\Delta g$ and $\Delta\gamma_{\phi}$ extracted randomly
for a finite length experiment – from Eqn. 61 by diagonalizing $M^{-1}$ and
expressing the elements of $\textbf{X}_{s}$ in terms of independent variables,
which can be easily sampled 444This 2-variate complex Gaussian distribution
can be equivalently sampled by extracting real and imaginary parts of
$\textbf{X}_{s}$ from a 4-variate (real) Gaussian distribution [50]., then
$\hat{\textbf{S}}$ is calculated through Eqn. 58. Once we have distributions
for $\hat{\textbf{S}}$, we calculate the $\pm\sigma$ intervals for
$\hat{S}_{\Delta g}$, $\hat{S}_{\Delta\gamma_{\phi}}$, $\hat{S}_{\Delta
g,\Delta\gamma_{\phi}}$ at all frequencies, for each experiment. If the fit is
indeed ”good”, then we should find that $\approx 68$% of the experimental data
fall within the posterior predicted intervals. Considering the fit performed
at single frequencies, we find that $\approx 85$% of experimental data fall in
the predicted intervals; this may indicate that we are slightly over-fitting
by using 13 parameters at each frequency, with 12 experimental data (four
separate experiments with three data, with PSD estimates for $\Delta g$,
$\Delta\gamma_{\phi}$, and their cross-spectrum). With regard to the fit to
smooth functions instead, we find 68% of experimental data falling in the
posterior prediction: this result is consistent even when not considering the
whole data-set, for example when estimating the goodness of fit in a single
experiment, or only considering noise in the translational acceleration
$\Delta g$, indicating that the model is able to properly reproduce the
observations.

####  VI.2.2 Model of force noise from in-band fluctuations and data analysis
technique

The analysis of ”in-band” voltage fluctuations is carried out analogously to
the previous subsection. We report here a quick summary of the models used. We
have only two experiments, with $V_{TM}\approx 0\,$V and then with
$V_{TM}\approx 1\,$V, which we label again with index $q$. In general our
model for acceleration/torque PSDs and cross spectra can be written as in Eqn.
23

$\displaystyle S_{\Delta g}$ $\displaystyle=S_{\Delta
g}^{bg}+\left(\frac{\frac{\partial C_{X}}{\partial
x}V_{TM}}{M}\right)^{2}S_{\Delta\left(\Delta_{x}\right)}$ $\displaystyle
S_{\Delta\gamma_{\phi}}$
$\displaystyle=S_{\Delta\gamma_{\phi}}^{bg}+\left(\frac{\frac{\partial
C_{X}}{\partial\phi}V_{TM}}{I}\right)^{2}S_{\Delta\left(\Delta_{\phi}\right)}$
(70) $\displaystyle S_{\Delta g,\Delta\gamma_{\phi}}$ $\displaystyle=S_{\Delta
g,\Delta\gamma_{\phi}}^{bg}+\frac{\frac{\partial C_{X}}{\partial
x}\frac{\partial
C_{X}}{\partial\phi}V_{TM}^{2}}{IM}S_{\Delta\left(\Delta_{x}\right),\Delta\left(\Delta_{\phi}\right)}$

As described in the text, an estimate of $S_{\Delta(\Delta_{x})}$ can be
extracted by simply fitting the first line in Eqn. 70 to the observed
$S_{\Delta g}$ using $S_{\Delta(\Delta_{x})}$ and $S^{bg}_{\Delta g}$ as
parameters, and using the one dimensional version of likelihood (63). Same
holds for $S_{\Delta(\Delta_{\phi})}$ and the second line of Eqn. 70. This was
done at single frequencies and also fitting smooth functions: in the first
case, given the very small amount of periodograms available (therefore wide
PSD distributions), a lower cutoff was needed on $S_{\Delta(\Delta_{x})}$ and
$S_{\Delta(\Delta_{\phi})}$, fixed to the very conservative 1/100 thermal
limit of actuation circuits. In the hypothesis of stray electrostatics noise
dominated by uncorrelated fluctuations in the electrode actuation voltages, we
can summarize the PSD model in Eqn. 70, similarly to Eqn. 55, as

$\textbf{M}(f,q)=\textbf{v}^{T}(q)\textbf{S}_{\Delta(\Delta)}(f)\textbf{v}(q)+\textbf{S}^{bg}(f)\,$
(71)

where $\textbf{S}^{bg}(f)$ is the same as that in Eqn. 55, while

$\displaystyle\textbf{v}(q)=V_{TM}(q)\begin{bmatrix}[c]\frac{\partial
C_{X}}{\partial x}/M&\frac{\partial C_{X}}{\partial\phi}/I\\\ -\frac{\partial
C_{X}}{\partial x}/M&\frac{\partial C_{X}}{\partial\phi}/I&\end{bmatrix}$
$\displaystyle\textbf{S}_{\Delta(\Delta)}(f)=\begin{bmatrix}S_{\Delta(\Delta_{14})}(f)&0\\\
0&S_{\Delta(\Delta_{23})}(f)\end{bmatrix}$ (72)

Finally, consistently with the approach for gain noise, we again use the
hypothesis of “statistically identical” electrodes, writing

$\displaystyle S_{\Delta(\Delta_{14})}=S_{\Delta}(1+\mu_{b})$ $\displaystyle
S_{\Delta(\Delta_{23})}=S_{\Delta}(1-\mu_{b})$ (73)

At each minimally correlated frequency, we use the same MCMC described in the
previous section with likelihood (Eqn. 63) to estimate the distributions of
the parameters $S_{\Delta}$, $S^{bg}_{\Delta g}$,
$S^{bg}_{\Delta\gamma_{\phi}}$ with a prior uniform in the logarithm, while
$\mu_{b}$ and $\xi$ (expressing the correlation between backgrounds as in Eqn.
55) with a uniform prior distribution across the interval $[-1,1]$.

We also perform a fit describing the relevant quantities as smooth functions
of frequency. While backgrounds are written as in (68), we write the other
quantities as

$\displaystyle
S_{\Delta(\Delta_{14})}=\frac{\epsilon_{14}}{f^{2}}+\frac{\rho_{14}}{f}+\eta_{14}$
$\displaystyle
S_{\Delta(\Delta_{23})}=\frac{\epsilon_{23}}{f^{2}}+\frac{\rho_{23}}{f}+\eta_{23}$
(74)

and MC estimate $\epsilon,\,\rho,\,\eta$ after a a re-parametrization similar
to Eqn. 73. The results indicate that only the $1/f$ component is
significantly different from zero.

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# All Prime Knots Through 10 Crossings Have Superbridge Index $\leq 5$

Clayton Shonkwiler  Department of Mathematics, Colorado State University, Fort
Collins, CO

###### Abstract

This paper gives new upper bounds on the stick numbers of the knots $9_{18}$,
$10_{18}$, $10_{58}$, $10_{66}$, $10_{68}$, $10_{80}$, $10_{82}$, $10_{84}$,
$10_{93}$, $10_{100}$, and $10_{152}$, as well as on the equilateral stick
number of $10_{79}$. These bounds imply that the knots $10_{58}$, $10_{66}$,
and $10_{80}$ have superbridge index $\leq 5$, completing the project of
showing that no prime knots through 10 crossings can have superbridge index
larger than 5. The current best bounds on stick number and superbridge index
for prime knots through 10 crossings are given in Appendix A.

The stick number and the superbridge index are two geometric knot invariants
which have proven quite difficult to compute. For example, the stick number is
known for only 35 of the 249 nontrivial knots through 10 crossings, and
superbridge index for only 49 [16, 9, 28].

The _stick number_ $\operatorname{stick}[K]$ of a knot type $K$ is the minimum
number of edges needed to construct a polygonal realization of $K$ [24], and
the _equilateral stick number_ $\operatorname{eqstick}[K]$ is the minimum
number of edges needed when we require all edges of the polygonal realization
to have the same length. These are elementary invariants of knots which give
some measure of the geometric (rather than purely topological) complexity of a
knot.

Another elementary invariant of knots is the superbridge index introduced by
Kuiper [15]. The _superbridge number_ $\operatorname{sb}(\gamma)$ of a (tamely
embedded) closed curve $\gamma$ is the maximum number of local maxima of any
projection of $\gamma$ to a line, and the _superbridge index_
$\operatorname{sb}[K]$ of a knot type $K$ is the minimum superbridge number of
any realization of $K$.

The key relationship between stick number and superbridge index is due to Jin
[14]: $\operatorname{sb}[K]\leq\frac{1}{2}\operatorname{stick}[K]$, because
there can’t be more critical points in any projection of a polygonal curve to
a line than there are vertices along the curve.

While there are a few general bounds on stick number [23, 12, 4, 11], the
typical way of producing upper bounds on the stick number of a particular knot
type is by finding examples [17, 18, 19, 6, 26, 25, 8, 9]. In turn, such a
bound on stick number provides, via Jin’s result, a bound on superbridge
index, though in general this bound can be arbitrarily bad: while there are
only finitely many knots $K$ with $\operatorname{stick}[K]\leq n$ for any $n$
[4, 23] (for example, there are exactly 11 knots with stick number $\leq 8$
[3, 5]), there are infinitely many knots with superbridge index equal to 4
(including all $(2,p)$-torus knots [15]).

The goal here is to provide new upper bounds on the stick numbers of 12 of the
9- and 10-crossing knots:

(a) $9_{18}$ (b) $10_{79}$

Figure 1: An equilateral 10-stick $9_{18}$ and an equilateral 11-stick
$10_{79}$. The vertex coordinates are given in Appendix B, and the knots are
shown in orthographic perspective, viewed from the direction of the positive
$z$-axis.

######  Theorem 1.

The knots $9_{18}$, $10_{18}$, $10_{68}$, $10_{82}$, $10_{84}$, $10_{93}$,
$10_{100}$, and $10_{152}$ have stick number and equilateral stick number
$\leq 10$ and the knots $10_{58}$, $10_{66}$, $10_{79}$, and $10_{80}$ have
stick number and equilateral stick number $\leq 11$.

###### Proof.

The result follows by exhibiting polygonal examples of each of the knots with
equal-length edges. Coordinates of these realizations are given in Appendix B
along with pictures of each knot; the pictures of the 10-stick $9_{18}$ and
the 11-stick $10_{79}$ are reproduced in Figure 1. These examples were found
among 130 billion random 10- and 11-gons generated in tight confinement using
the algorithms and code from [9, 7]; this took approximately 35,000 core-hours
on a quad-core machine with Intel Xeon E5 processors. More complete analysis
of this data will appear elsewhere.

Since these examples are represented by the double-precision floating point
coordinates of their vertices, they are only approximately equilateral.
However, a result of Millett and Rawdon [21] guarantees the existence of a
true equilateral realization of a knot provided the edgelengths
$L_{1},\dots,L_{n}\approx 1$ and minimum distance $\mu$ between non-adjacent
edges in an approximately equilateral realization satisfy

$|L_{i}-1|<\min\left\\{\frac{\mu}{n},\frac{\mu^{2}}{4}\right\\}$

for all $i=1,\dots,n$. For all of the knots recorded in Appendix B we have

$|L_{i}-1|<10^{-2.96}\min\left\\{\frac{\mu}{n},\frac{\mu^{2}}{4}\right\\},$

so the Millett–Rawdon criterion is easily satisfied and these examples also
provide upper bounds on equilateral stick number. ∎

For all knots mentioned in the theorem except $10_{79}$, the stick number
bound is new. It was already known [26, 27] that
$\operatorname{stick}[10_{79}]\leq 11$, but the previous best bound on
equilateral stick number was $12$ [25]. Since the known bounds on stick number
and equilateral stick number for this knot now agree, it can be removed from
Rawdon and Scharein’s [25] list of candidates for knots with unequal stick
number and equilateral stick number. Indeed, combined with prior results [9,
20], this implies that the only knot remaining from that list is $9_{29}$,
which is known to have stick number 9 [26, 27], but for which the best upper
bound on equilateral stick number is 10. Removing $10_{79}$ from the list
would seem to slightly weaken the evidence that stick number and equilateral
stick number are distinct invariants, but it remains implausible to think that
these two invariants will always agree.

Using Jin’s bound
$\operatorname{sb}[K]\leq\frac{1}{2}\operatorname{stick}[K]$, Theorem 1
implies that all of the knots mentioned have superbridge index $\leq 5$. This
result is new for the knots $10_{58}$, $10_{66}$, and $10_{80}$:

######  Corollary 2.

The knots $10_{58}$, $10_{66}$, and $10_{80}$ have superbridge index $\leq 5$.
In particular, this implies that all prime knots through 10 crossings have
superbridge index $\leq 5$.

###### Proof.

As mentioned, the first sentence is an immediate consequence of Theorem 1 and
Jin’s bound. The second sentence follows because all other 10-crossing knots
were previously known to have superbridge index $\leq 5$. Indeed, Theorem 1
implies that, with at most two possible exceptions, all 10-crossing knots have
stick number $\leq 11$. The two exceptions are $10_{37}$ and $10_{76}$, both
of which have been realized with 12 sticks [25, 9]. While Jin’s bound does not
imply these knots have superbridge index $\leq 5$, such a bound has recently
been proved (by very different methods) for both knots: since $10_{37}$ is a
2-bridge knot, the result $\operatorname{sb}[10_{37}]\leq 5$ is a consequence
of the bound $\operatorname{sb}[K]\leq 3\operatorname{b}[K]-1$ proved by Adams
et al. [1], where $\operatorname{b}[K]$ is the bridge index of the knot
$K$;111For completeness, an explicit realization of $10_{37}$ with superbridge
number equal to 5 is given in Appendix C. the result
$\operatorname{sb}[10_{76}]\leq 5$ was proved in prior work [28] by finding an
explicit 12-stick realization of $10_{76}$ with superbridge number equal to 5.
∎

Corollary 2, together with what was already known about the superbridge index
of knots through 10 crossings (as recorded in Appendix A), provides modest
evidence for the following conjecture:

######  Conjecture 3.

For a knot $K$ with crossing number $\operatorname{cr}[K]\geq 7$,

$\operatorname{sb}[K]\leq\left\lceil\frac{\operatorname{cr}[K]}{2}\right\rceil.$
(1)

This is somewhat stronger than the known bound
$\operatorname{sb}[K]\leq\frac{3}{4}(\operatorname{cr}[K]+1)$, which follows
from Jin’s bound and Huh and Oh’s [12] result
$\operatorname{stick}[K]\leq\frac{3}{2}(\operatorname{cr}[K]+1)$.

3 holds for the one infinite family of knots for which superbridge index is
known, namely the torus knots $T_{p,q}$. Assuming $2\leq p<q$, Kuiper [15]
proved that $\operatorname{sb}[T_{p,q}]=\min\\{2p,q\\}$, Murasugi [22] showed
that $\operatorname{cr}[T_{p,q}]=q(p-1)$, and it is straightforward to verify
that these quantities satisfy (1) so long as
$7\leq\operatorname{cr}[T_{p,q}]=q(p-1)$.

## Acknowledgments

Thanks, as always, to Jason Cantarella and Tom Eddy for their mathematical and
computational knowledge and to Allison Moore and Chuck Livingston for
maintaining KnotInfo [16]. This work was partially supported by grants from
the Simons Foundation (#354225 and #709150).

##  Appendix A Stick Number and Superbridge Index Bounds

Bounds on stick number and superbridge index for prime knots through 10
crossings are given below, including references for where these results were
proved. If an exact value is not known, the possible values, as determined by
known upper and lower bounds, are given in the form of an interval; e.g., the
entry $[9,11]$ for $\operatorname{stick}[9_{6}]$ means that
$9\leq\operatorname{stick}[9_{6}]\leq 11$.

The citations given in the table are mostly references for the upper bounds,
since the lower bounds always come from the same sources: for unknown stick
numbers the lower bound always comes from Calvo’s characterization of knots
with stick number $\leq 8$ [3, 5] and the lower bound on superbridge index
always comes from either Kuiper’s result
$\operatorname{b}[K]<\operatorname{sb}[K]$ for nontrivial knots [15] or Jeon
and Jin’s characterization of possible 3-superbridge knots [13].

The bounds for stick number and equilateral stick number agree for all knots
in the table except $9_{29}$, which has $\operatorname{stick}[9_{29}]=9$ and
$\operatorname{eqstick}[9_{29}]\leq 10$. This stick number is marked by an
asterisk in the table as a reminder.

$K$ | $\operatorname{stick}[K]$ | $\operatorname{sb}[K]$ |
---|---|---|---
$0_{1}$ | 3 | 1 |
$3_{1}$ | 6 | 3 | [24, 15]
$4_{1}$ | 7 | 3 | [14, 24]
$5_{1}$ | 8 | 4 | [23, 17, 15]
$5_{2}$ | 8 | $[3,4]$ | [14]
$6_{1}$ | 8 | $[3,4]$ | [23, 17, 14]
$6_{2}$ | 8 | $[3,4]$ | [23, 17, 14]
$6_{3}$ | 8 | $[3,4]$ | [23, 17, 14]
$7_{1}$ | 9 | 4 | [15]
$7_{2}$ | 9 | $[3,4]$ | [17, 14]
$7_{3}$ | 9 | $[3,4]$ | [17, 14]
$7_{4}$ | 9 | $[3,4]$ | [17, 14]
$7_{5}$ | 9 | 4 | [17, 14]
$7_{6}$ | 9 | 4 | [17, 14]
$7_{7}$ | 9 | 4 | [17, 14]
$8_{1}$ | $[9,10]$ | 4 | [25, 28]
$8_{2}$ | $[9,10]$ | 4 | [25, 28]
$8_{3}$ | $[9,10]$ | 4 | [25, 28]
$8_{4}$ | $[9,10]$ | $[3,4]$ | [25, 28]
$8_{5}$ | $[9,10]$ | 4 | [25, 28]
$8_{6}$ | $[9,10]$ | 4 | [25, 28]
$8_{7}$ | $[9,10]$ | 4 | [25, 28]
$8_{8}$ | $[9,10]$ | 4 | [25, 28]
$8_{9}$ | $[9,10]$ | $[3,4]$ | [25, 28]
$8_{10}$ | $[9,10]$ | 4 | [25, 28]
$8_{11}$ | $[9,10]$ | 4 | [25, 28]
$8_{12}$ | $[9,10]$ | 4 | [25, 28]
$8_{13}$ | $[9,10]$ | 4 | [25, 28]
$K$ | $\operatorname{stick}[K]$ | $\operatorname{sb}[K]$ |
---|---|---|---
$8_{14}$ | $[9,10]$ | 4 | [25, 28]
$8_{15}$ | $[9,10]$ | 4 | [25, 28]
$8_{16}$ | 9 | 4 | [25, 14]
$8_{17}$ | 9 | 4 | [25, 14]
$8_{18}$ | 9 | 4 | [4, 25, 14]
$8_{19}$ | 8 | 4 | [2, 14, 20, 15]
$8_{20}$ | 8 | 4 | [23, 17, 14]
$8_{21}$ | 9 | 4 | [17, 14]
$9_{1}$ | $[9,10]$ | 4 | [25, 15]
$9_{2}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$9_{3}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$9_{4}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{5}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{6}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$9_{7}$ | $[9,10]$ | 4 | [25, 28]
$9_{8}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{9}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{10}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{11}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$9_{12}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{13}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{14}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{15}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$9_{16}$ | $[9,10]$ | 4 | [25, 28]
$9_{17}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{18}$ | $[9,10]$ | $[4,5]$ |  Theorem 1, [14]
$9_{19}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{20}$ | $[9,10]$ | 4 | [25, 28]

$K$ | $\operatorname{stick}[K]$ | $\operatorname{sb}[K]$ |
---|---|---|---
$9_{21}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$9_{22}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{23}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$9_{24}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{25}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$9_{26}$ | $[9,10]$ | 4 | [25, 28]
$9_{27}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$9_{28}$ | $[9,10]$ | 4 | [25, 28]
$9_{29}$ | 9∗ | 4 | [26, 14]
$9_{30}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{31}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{32}$ | $[9,10]$ | 4 | [25, 28]
$9_{33}$ | $[9,10]$ | 4 | [25, 28]
$9_{34}$ | 9 | 4 | [25, 14]
$9_{35}$ | 9 | 4 | [9, 14]
$9_{36}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$9_{37}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{38}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$9_{39}$ | 9 | 4 | [9, 14]
$9_{40}$ | 9 | 4 | [26, 14]
$9_{41}$ | 9 | 4 | [26, 14]
$9_{42}$ | 9 | 4 | [6, 14]
$9_{43}$ | 9 | 4 | [9, 14]
$9_{44}$ | 9 | 4 | [25, 14]
$9_{45}$ | 9 | 4 | [9, 14]
$9_{46}$ | 9 | 4 | [6, 14]
$9_{47}$ | 9 | 4 | [25, 14]
$9_{48}$ | 9 | 4 | [9, 14]
$9_{49}$ | 9 | 4 | [25, 14]
$10_{1}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{2}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{3}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{4}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{5}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{6}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{7}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{8}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{9}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{10}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{11}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{12}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{13}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{14}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{15}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{16}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{17}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{18}$ | $[9,10]$ | $[4,5]$ |  Theorem 1, [14]
$10_{19}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{20}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{21}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{22}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{23}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{24}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{25}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{26}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{27}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{28}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{29}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{30}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{31}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{32}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{33}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{34}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{35}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{36}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{37}$ | $[9,12]$ | $[4,5]$ | [25, 1]
$10_{38}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{39}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{40}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{41}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{42}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{43}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{44}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{45}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{46}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{47}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{48}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{49}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{50}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{51}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{52}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{53}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{54}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{55}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{56}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{57}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{58}$ | $[9,11]$ | $[4,5]$ |  Theorem 1, Corollary 2
$10_{59}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{60}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{61}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{62}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{63}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{64}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{65}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{66}$ | $[9,11]$ | $[4,5]$ |  Theorem 1, Corollary 2
$10_{67}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{68}$ | $[9,10]$ | $[4,5]$ |  Theorem 1, [14]
$10_{69}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{70}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{71}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{72}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{73}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{74}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{75}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{76}$ | $[9,12]$ | $[4,5]$ | [9, 28]
$10_{77}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{78}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{79}$ | $[9,11]$ | $[4,5]$ | [14, 26]
$10_{80}$ | $[9,11]$ | $[4,5]$ |  Theorem 1, Corollary 2
$10_{81}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{82}$ | $[9,10]$ | $[4,5]$ |  Theorem 1, [14]
$10_{83}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{84}$ | $[9,10]$ | $[4,5]$ |  Theorem 1, [14]
$10_{85}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{86}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{87}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{88}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{89}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{90}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{91}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{92}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{93}$ | $[9,10]$ | $[4,5]$ |  Theorem 1, [14]
$10_{94}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{95}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{96}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{97}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{98}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{99}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{100}$ | $[9,10]$ | $[4,5]$ |  Theorem 1, [14]
$10_{101}$ | $[9,11]$ | $[4,5]$ | [9, 14]
$10_{102}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{103}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{104}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{105}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{106}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{107}$ | $[9,10]$ | $[4,5]$ | [26, 14]
$10_{108}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{109}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{110}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{111}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{112}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{113}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{114}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{115}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{116}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{117}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{118}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{119}$ | $[9,10]$ | $[4,5]$ | [26, 14]
$10_{120}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{121}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{122}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{123}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{124}$ | 10 | 5 | [2, 14, 15]
$10_{125}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{126}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{127}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{128}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{129}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{130}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{131}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{132}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{133}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{134}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{135}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{136}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{137}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{138}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{139}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{140}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{141}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{142}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{143}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{144}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{145}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{146}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{147}$ | $[9,10]$ | $[4,5]$ | [26, 14]
$10_{148}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{149}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{150}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{151}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{152}$ | $[9,10]$ | $[4,5]$ |  Theorem 1, [14]
$10_{153}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{154}$ | $[9,11]$ | $[4,5]$ | [25, 14]
$10_{155}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{156}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{157}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{158}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{159}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{160}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{161}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{162}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{163}$ | $[9,10]$ | $[4,5]$ | [25, 14]
$10_{164}$ | $[9,10]$ | $[4,5]$ | [9, 14]
$10_{165}$ | $[9,10]$ | $[4,5]$ | [25, 14]

##  Appendix B Knot Images and Coordinates

The knot coordinates are normalized so that the first vertex is at the origin,
the second is at $(1,0,0)$, and the third is in the $xy$-plane with positive
$y$-coordinate. Each knot is shown in orthographic perspective from the
direction of the positive $z$-axis. These coordinates can also be downloaded
from the stick-knot-gen project [7].

$\boldmath{9_{18}}$

0. | 0. | 0.
---|---|---
0.9999999999999999 | 0. | 0.
0.1369811385433643 | 0.5051716983067165 | 0.
0.36445473772481973 | -0.46837424343037093 | 0.02154207517120799
-0.40824166622547475 | -0.1946531623448116 | 0.5942697915353068
0.1761863718304549 | 0.05044370765670344 | -0.17927490184029685
-0.0358214851573142 | 0.11220243094991508 | 0.796039677499439
0.36991848186843224 | 0.4169202041594481 | -0.065657576672912
-0.2483842722893944 | -0.23075441407147784 | 0.3795606766593944
0.6123146506430659 | -0.7330077964576398 | 0.29626059295858787

$\boldmath{10_{18}}$

0. | 0. | 0.
---|---|---
0.9999999999999999 | 0. | 0.
0.10374904044780832 | 0.4435473114581754 | 0.
0.4061269411819969 | -0.1391324357591598 | 0.7543552991331917
0.05220407842345304 | 0.41171347034152267 | -0.0014934919281650513
1.0010147153854116 | 0.09608809195992474 | 0.01029615222744267
0.06703161095626266 | -0.2530860092405562 | -0.06555243175225965
0.007156648478157002 | 0.6766297479925333 | 0.29782560778886097
0.6758185737598675 | 0.07818217741767258 | -0.14348124118541358
0.5092073572964613 | 0.21798840253364535 | 0.8325796800522959

$\boldmath{10_{58}}$

0. | 0. | 0.
---|---|---
0.9999999999999999 | 0. | 0.
0.23075419502569852 | 0.6389529650368944 | 0.
0.34568868227993477 | -0.34762407938349305 | 0.1159991338861572
0.5854402278388353 | 0.6046839598660042 | 0.304754517340236
-0.34288871608553806 | 0.39659016855951384 | -0.00330772619497593
0.5747045141992497 | 0.1278600084591875 | 0.28962019119765003
-0.19231721046439432 | -0.13517667062917393 | -0.2956057304327706
0.3810463453997557 | 0.2597856620372011 | 0.4222096975022616
0.13411123197649716 | 0.5946070064410464 | -0.487140464476194
0.7640898592707 | 0.5766222162271076 | 0.2892637321079513

$\boldmath{10_{66}}$

0. | 0. | 0.
---|---|---
0.9999999999999999 | 0. | 0.
0.5145101891023756 | 0.8742423253964475 | 0.
0.5275451799672034 | -0.12484964472859603 | -0.040562596623153806
-0.13361019429763404 | 0.5339205870767882 | 0.318457392747638
0.3118131683954475 | -0.3000760437290037 | 0.6441069709321883
0.5621860061072727 | 0.3292462605214777 | -0.09160144923778166
-0.24397323845573143 | -0.1795895608670905 | 0.21038096627250544
0.5513339449909647 | 0.2862276622235819 | -0.17756335786457722
-0.17814186103165766 | 0.12471725485537188 | 0.4871015477377872
0.8009432011351053 | 0.32768567714920493 | 0.5011108515555394

$\boldmath{10_{68}}$

0. | 0. | 0.
---|---|---
0.9999999999999999 | 0. | 0.
0.14308910801988475 | 0.5154645702721413 | 0.
0.5507160581438794 | -0.3950793595184551 | 0.0689203994047325
-0.220985520662934 | 0.2155209342031813 | -0.10896707787830442
0.7019924653376205 | 0.21890712528211487 | -0.4938049282365256
0.014394765753044353 | 0.5527134207336278 | 0.1510072619731009
0.059021448275970674 | -0.38897111542175017 | -0.18251739014848975
0.5697203717471668 | 0.36984460712635464 | 0.2216920961072869
0.6359432500426695 | 0.16273761313218546 | -0.7543822983058428

$\boldmath{10_{79}}$

0. | 0. | 0.
---|---|---
0.9999999999999999 | 0. | 0.
0.15662097603563974 | 0.537319106245928 | 0.
0.2445856042154266 | -0.43538593154418503 | -0.21472571724627276
0.6450217925958391 | 0.1626522315261948 | 0.4795373557647955
0.734006748172326 | 0.23463544410470605 | -0.5138910990259495
0.36493154829639873 | 0.47001886661877124 | 0.3852074168454255
0.021330163575223243 | 0.10403072835102278 | -0.47965718010891734
0.3179757846004717 | -0.018939867230503504 | 0.46738009906697064
0.4835968181650272 | -0.48616249117625676 | -0.401108641743325
0.7849150964041073 | 0.4673440590915645 | -0.40681423508617587

$\boldmath{10_{80}}$

0. | 0. | 0.
---|---|---
0.9999999999999999 | 0. | 0.
0.3327312868213165 | 0.7448170677507757 | 0.
0.41164114869381774 | -0.18477978499933626 | 0.36003183894809804
-0.37657987371245755 | 0.35524645381103787 | 0.06493985119999454
0.4472049020942717 | -0.03782867814303159 | -0.3435581093681675
0.44180933980825615 | 0.7428150212139177 | 0.2813949309171927
-0.3132726693922105 | 0.1457679277099991 | 0.010496559714429563
0.5497691607609388 | 0.36988149308037604 | -0.4421974972632168
0.19881030933734742 | -0.13460231801890482 | 0.3466776316045772
0.8574892644307307 | 0.5132409345720769 | -0.03599867310924573

$\boldmath{10_{82}}$

0. | 0. | 0.
---|---|---
0.9999999999999999 | 0. | 0.
0.09089527821440953 | 0.41656764736011836 | 0.
0.9122855066438609 | 0.04664743439303293 | -0.4341395267411148
0.4293464164746975 | -0.5569166222908565 | 0.20027379646849952
0.8672055548188717 | 0.28186943587667174 | -0.12332676532377572
0.2256114315412598 | -0.4239913296650354 | -0.4235226357892222
0.901013770664066 | -0.17005456734157295 | 0.2688268431938411
0.651739366157114 | 0.46199768313726536 | -0.4649116823875408
0.8761992708919832 | -0.45907319072845804 | -0.14671960756064373

$\boldmath{10_{84}}$

0. | 0. | 0.
---|---|---
0.9999999999999999 | 0. | 0.
0.19130320128681053 | 0.5882257115691553 | 0.
0.18504701394830536 | -0.36160278132829377 | 0.3127086410704792
0.4007142236869289 | 0.4473066686279016 | -0.23424031122317077
0.6920922472558773 | 0.29033346909917157 | 0.7094006491058975
0.6101485945447912 | -0.17959763305314008 | -0.1694906962891281
-0.10778017173660057 | 0.06099088207608897 | 0.483728602976893
0.5896371143947019 | -0.27762434838744 | -0.14789537907522962
0.7331230152034498 | 0.6547968310753483 | 0.1837709296724706

$\boldmath{10_{93}}$

0. | 0. | 0.
---|---|---
0.9999999999999999 | 0. | 0.
0.4322831520544033 | 0.8232238945503928 | 0.
0.366749435669193 | -0.16760413688857084 | -0.11817421094101156
0.6013038771595937 | 0.7421884288832536 | 0.22426074820224834
0.8484129178261877 | 0.12369596771944935 | -0.5216643133852914
0.1699818416631752 | -0.08145086977191296 | 0.18377601007487826
0.5088788663970447 | 0.3280505164135769 | -0.6632525734680226
0.23773536251973704 | 0.8273614792778586 | 0.1596506788458079
0.8793957949879448 | 0.28424997300173205 | -0.3819227521450315

$\boldmath{10_{100}}$

0. | 0. | 0.
---|---|---
0.9999999999999999 | 0. | 0.
0.25025317408203634 | 0.6617247894902673 | 0.
0.08590073137649616 | -0.2865914931622944 | -0.2714488950663621
0.225196070444289 | 0.328672653046371 | 0.5044678781149572
0.4853999222371724 | 0.08756968372166599 | -0.43049913397421674
0.6948680476379834 | 0.6204316823737933 | 0.3893674373857395
-0.03774921617740326 | 0.5293732483746156 | -0.28515484271868646
0.7101850493261617 | -0.060296929991126846 | 0.01961296271974253
0.4375512255320434 | 0.8835485041661305 | -0.16700588558850374

$\boldmath{10_{152}}$

0. | 0. | 0.
---|---|---
0.9999999999999999 | 0. | 0.
0.23742406673458422 | 0.6468987138682376 | 0.
0.8624816316358974 | -0.04229570782933964 | -0.36648886703497885
0.4011057049003684 | 0.5864895424110548 | 0.2594195634155591
-0.1801631109472681 | -0.1253414478506597 | -0.13481793483405285
0.6601561286258801 | 0.38286488530589013 | 0.0538349170998087
-0.008962871699621913 | 0.35627600812192406 | -0.6888445528019091
0.09031617352847202 | -0.09176640719170803 | 0.1996382355842952
0.9044534650000824 | 0.4199643332833877 | -0.07479230187114154

##  Appendix C A 5-Superbridge Realization of $10_{37}$

Since $10_{37}$ is a 2-bridge knot and Adams et al. [1] showed that
$\operatorname{sb}[K]\leq 3\operatorname{b}[K]-1$, it follows that
$\operatorname{sb}[10_{37}]\leq 5$, even though the best known bound on stick
number is $\operatorname{stick}[10_{37}]\leq 12$ [25]. Adams et al.’s proof is
based on putting the knot in a standard configuration that has superbridge
number $3\operatorname{b}[K]-1$, so one could in principle follow their
procedure to find a concrete realization of $10_{37}$ with superbridge number
equal to 5.

Rather than doing that, here is a $12$-stick realization of $10_{37}$ with
superbridge number 5:

$10_{37}$
---
|  $0$ |  $0$ |  $0$ |  $1$
|  $1000$ |  $0$ |  $0$ |  $1$
|  $156$ |  $536$ |  $0$ |  $1$
|  $655$ |  $-107$ |  $-580$ |  $1$
|  $779$ |  $-509$ |  $327$ |  $1$
|  $73$ |  $-290$ |  $-347$ |  $20889658716$
|  $903$ |  $-312$ |  $211$ |  $1$
|  $557$ |  $-322$ |  $-728$ |  $1$
|  $592$ |  $362$ |  $1$ |  $637602926$
|  $521$ |  $-622$ |  $162$ |  $1683870031$
|  $381$ |  $227$ |  $-347$ |  $116253911$
|  $847$ |  $80$ |  $525$ |  $20181547124$

The knot is shown in orthographic perspective from the direction of the
positive $z$-axis relative to the vertex coordinates given in the three
columns to the right of the image. These coordinates have been rounded to
three significant digits and expressed as integers so that they are amenable
to exact arithmetic. The last column gives the coordinates of a vector
certifying that this specific $10_{37}$ has superbridge number $\leq 5$, using
the approach from [28].

Specifically, a 12-stick realization of a knot $K$ has superbridge number
equal to 6 only if there exists a direction $\vec{v}\in S^{2}$ so that the
projection of each of the 12 vertices to the line spanned by $\vec{v}$ is
either a local minimum or a local maximum. If $\vec{e}_{i}$ is the $i$th edge
vector, this means we can find $\vec{v}$ so that
$\vec{v}\cdot\vec{e}_{1},\dots,\vec{v}\cdot\vec{e}_{12}$ alternates signs, or
(after possibly replacing $\vec{v}$ with $-\vec{v}$) equivalently

$\vec{v}^{T}\begin{bmatrix}\vec{e}_{1}&-\vec{e}_{2}&\cdots&\vec{e}_{11}&-\vec{e}_{12}\end{bmatrix}$

has all positive entries. By Gordan’s theorem [10], either such a $\vec{v}$
exists, or there exists $\vec{u}\in\mathbb{R}^{12}$ such that

$\begin{bmatrix}\vec{e}_{1}&-\vec{e}_{2}&\cdots&\vec{e}_{11}&-\vec{e}_{12}\end{bmatrix}\vec{u}=\vec{0},$

but not both. The last column above gives such a $\vec{u}$ (found using
_Mathematica_ ’s FindInstance function), which provides a certificate that the
superbridge number of this realization of $10_{37}$ is strictly less than 6.

## References

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  * [11] Youngsik Huh, Sungjong No, and Seungsang Oh.  Stick numbers of 2-bridge knots and links.  Proceedings of the American Mathematical Society, 139(11):4143–4152, 2011.
  * [12] Youngsik Huh and Seungsang Oh.  An upper bound on stick number of knots.  Journal of Knot Theory and Its Ramifications, 20(5):741–747, 2011\.
  * [13] Choon Bae Jeon and Gyo Taek Jin.  A computation of superbridge index of knots.  Journal of Knot Theory and Its Ramifications, 11(3):461–473, 2002\.
  * [14] Gyo Taek Jin.  Polygon indices and superbridge indices of torus knots and links.  Journal of Knot Theory and Its Ramifications, 6(2):281–289, 1997\.
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  * [16] Charles Livingston and Allison H Moore.  KnotInfo: Table of Knot Invariants.  https://knotinfo.math.indiana.edu.
  * [17] Monica Meissen.  Edge number results for piecewise-linear knots.  In Vaughan F R Jones, Joanna Kania-Bartoszyńska, Józef H Przytycki, Paweł Traczyk, and Vladimir G Turaev, editors, Knot Theory: Papers from the Mini-Semester Held in Warsaw, July 13–August 17, 1995, volume 42 of Banach Center Publications, pages 235–242. Polish Academy of Sciences, Institute of Mathematics, Warsaw, 1998.
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  * [20] Kenneth C Millett.  Physical knot theory: An introduction to the study of the influence of knotting on the spatial characteristics of polymers.  In Louis H Kauffman, Sofia Lambropoulou, Slavik Jablan, and Józef H Przytycki, editors, Introductory Lectures on Knot Theory, volume 46 of Series on Knots and Everything, pages 346–378. World Scientific Publishing, Singapore, 2012.
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  * [22] Kunio Murasugi.  On the braid index of alternating links.  Transactions of the American Mathematical Society, 326(1):237–260, 1991.
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# A Note on the Continuity of Expected Utility Functions

Yuhki Hosoya
Faculty of Economics, Chuo University TEL: +81-90-5525-5142, E-mail:
ukki(at)gs.econ.keio.ac.jp742-1, Higashinakano, Hachioji-shi, Tokyo, 192-0393,
Japan.

###### Abstract

In this paper, we study the continuity of expected utility functions, and
derive a necessary and sufficient condition for a weak order on the space of
simple probabilities to have a continuous expected utility function. We also
verify that almost the same condition is necessary and sufficient for a weak
order on the space of probabilities with compact-support to have a continuous
expected utility function.

Keywords: Expected Utility, Continuity, Compact-Support Probability,
Sequential Continuity.

MSC2020 code: 91B06, 91A30.

##  1 Introduction

von Neumann and Morgenstern (1944) presented a necessary and sufficient
condition for a weak order on the space of simple probability measures on a
set $X$ to be represented in terms of the greater or lesser expected value of
a function $u:X\to\mathbb{R}$. This $u$ is called an expected utility
function. The expected utility function has a wide range of applications in
economics and other social sciences, and is still actively studied.

Neumann–Morgenstern’s result does not require any topology on the set $X$.
However, the most important application of this result is when $X$ is a half-
line in $\mathbb{R}$, which is naturally a topological space. It is natural
that the continuity of $u$ would be of interest. Therefore, several
researchers sought conditions for $u$ to be continuous. Grandmont (1972) is a
representative result in this research area. However, a serious problem
arises: if we try to introduce a natural condition for $u$ to be continuous,
$u$ becomes always bounded. When $X$ is a half-line, functions that we would
want to treat very naturally, such as $\sqrt{x}$ and $\log x$, are all
unbounded, which means that the usual result on the continuity of $u$ cannot
be treated in the application.

In this paper, we consider identifying a necessary and sufficient condition
for $u$ to be continuous. First, following Neumann–Morgenstern, we consider
the space of simple probability measures on $X$. If $X$ is Hausdorff, then
these are Borel probability measures. Under this setup, we derive a necessary
and sufficient condition for an expected utility function to be continuous.
(Theorem 1)

Next, we change the object of consideration to the space of compact-support
Borel probability measures. If $X$ is a separable metric space, then the space
of simple probability measures is dense in the whole space of Borel
probability measures under the weak* topology. Therefore, by a natural
extension of Theorem 1, we can still derive a necessary and sufficient
condition for an expected utility function to be continuous. (Theorem 2)

The structure of this paper is as follows. In Section 2, we define basic terms
and derive some basic results. Section 3 is devoted to the derivation of the
main results. In Section 4, we discuss the position of this study in
comparison with related studies.

##  2 Preliminaries

Consider that a nonempty set $X$ is given. A subset $\succsim$ of $X^{2}$ is
called a binary relation on $X$. A binary relation $\succsim$ on $X$ is called
a weak order on $X$ if and only if, 1) (completeness) either
$(x,y)\in\succsim$ or $(y,x)\in\succsim$ for every $x,y\in X$, and 2)
(transitivity) if $(x,y)\in\succsim$ and $(y,z)\in\succsim$, then
$(x,z)\in\succsim$. If $\succsim$ is a weak order on $X$, then we write
$x\succsim y$ instead of $(x,y)\in\succsim$. Moreover, we write $x\succ y$ if
and only if $x\succsim y$ and $y\not\succsim x$, and $x\sim y$ if and only if
$x\succsim y$ and $y\succsim x$.

Suppose that $\succsim$ is a weak order on $X$. If a function
$u:X\to\mathbb{R}$ satisfies the following relationship:

$x\succsim y\Leftrightarrow u(x)\geq u(y),$

then $u$ is said to represent $\succsim$.

Next, suppose that $X$ is a Hausdorff topological space. Suppose that $P$ is a
regular Borel probability measure on $X$.111We say that a Borel probability
measure $P$ is regular if and only if, for any Borel set $A$,
$P(A)=\sup\\{P(K)|K\mbox{ is compact},\ K\subset A\\}$. Then, it is known that
there exists a unique least closed set $F^{*}$ such that $P(F^{*})=1$.222Let
$F^{*}$ be the set of all $x\in X$ such that for any open neighborhood $U$ of
$x$, $P(U)>0$. We can easily show that $F^{*}$ is closed, and $P(X\setminus
F^{*})=0$ by the regularity. We call such an $F^{*}$ the support of $P$. If
the support is compact, then we call $P$ compact-support, and if the support
is finite, then we call $P$ simple.

Suppose that $X$ is a Hausdorff topological space, $P$ is a regular Borel
probability measure, and $u:X\to\mathbb{R}$ is a Borel measurable function.
Let $E_{P}[u]$ denote the expectation of $u$ with respect to $P$. If $P$ is
simple and $\\{x_{1},...,x_{m}\\}$ is the support of $P$, then

$E_{P}[u]=\sum_{i=1}^{m}P(\\{x_{i}\\})u(x_{i}).$

Let $\mathscr{P}_{X}$ be the set of all regular Borel probability measures,
$\mathscr{P}_{c}$ be the set of all compact-support elements in
$\mathscr{P}_{X}$, and $\mathscr{P}_{s}$ be the set of all simple elements in
$\mathscr{P}_{X}$. Note that, both $\mathscr{P}_{c}$ and $\mathscr{P}_{s}$ are
convex. Suppose that $Y\subset X$. Let $\mathscr{P}_{Y}$ denote the set of all
$P\in\mathscr{P}_{X}$ such that the support of $P$ is included in $Y$. Suppose
also that $\succsim$ is a weak order on $\mathscr{P}\subset\mathscr{P}_{X}$.
Then, define $\succsim_{Y}=\succsim\cap(\mathscr{P}_{Y})^{2}$. Note that,
$\succsim_{Y}$ is a weak order on $\mathscr{P}_{Y}\cap\mathscr{P}$.

Suppose that $\mathscr{P}$ is a convex subset of $\mathscr{P}_{X}$, and
$\succsim$ is a weak order on $\mathscr{P}$. Then, it is said to be
independent if and only if for every $P,Q\in\mathscr{P}$, $P\succ Q$ implies
that

$(1-t)P+tR\succ(1-t)Q+tR$

for every $R\in\mathscr{P}$ and every $t\in[0,1[$. Moreover, it is said to be
segmentally continuous if and only if for every $P,Q,R\in\mathscr{P}$ such
that $P\succ Q\succ R$, the following two sets

$\\{t\in[0,1]|(1-t)P+tR\succsim Q\\},\ \\{t\in[0,1]|Q\succsim(1-t)P+tR\\}$

are closed. If there exists a function $u:X\to\mathbb{R}$ such that
$U(P)\equiv E_{P}[u]$ represents $\succsim$, then we call $u$ a
Neumann–Morgenstern expected utility function, or abbreviately, an expected
utility function of $\succsim$.

We need the following lemmas.

Lemma 1. Suppose that $\mathscr{P}$ is a convex subset of $\mathscr{P}_{X}$
that includes $\mathscr{P}_{s}$, and $\succsim$ is a weak order on
$\mathscr{P}$. If $\succsim$ has an expected utility function, then $\succsim$
is independent. Moreover, such an expected utility function is unique up to a
positive affine transformation.

Proof. Suppose that there exists an expected utility function $u$ of
$\succsim$. Let $P,Q,R\in\mathscr{P}$ and $P\succ Q$. Then, for any
$t\in[0,1[$,

$(1-t)E_{P}[u]+tE_{R}[u]>(1-t)E_{Q}[u]+tE_{R}[u],$

which implies that $(1-t)P+tR\succ(1-t)Q+tR$. Therefore, $\succsim$ is
independent.

Suppose that $v$ is another expected utility function of $\succsim$. If $u$ is
a constant function, then $v$ is also a constant function, and thus $v$ is a
positive affine transformation of $u$. Hence, we assume that there exists
$x^{*},y^{*}\in X$ such that $u(x^{*})>u(y^{*})$. Without loss of generality,
we assume that $u(x^{*})=1,u(y^{*})=0$.

Let $\delta_{x}$ denote the Dirac measure whose support is $\\{x\\}$. Choose
any $x\in X$. Then,

$u(x)=u(x)u(x^{*})+(1-u(x))u(y^{*}).$

If $u(x)\in[0,1]$, then $\delta_{x}\sim
u(x)\delta_{x^{*}}+(1-u(x))\delta_{y^{*}}$, and thus
$v(x)=(v(x^{*})-v(y^{*}))u(x)+v(y^{*})$. If $u(x)>1$, then
$\delta_{x^{*}}\sim\frac{1}{u(x)}\delta_{x}+\frac{u(x)-1}{u(x)}\delta_{y^{*}}$,
and thus $v(x^{*})=\frac{v(x)-v(y^{*})}{u(x)}+v(y^{*})$, which implies that
$v(x)=(v(x^{*})-v(y^{*}))u(x)+v(y^{*})$. If $u(x)<0$, then
$\delta_{y^{*}}\sim\frac{-u(x)}{1-u(x)}\delta_{x^{*}}+\frac{1}{1-u(x)}\delta_{x}$,
and thus $v(y^{*})=-\frac{u(x)}{1-u(x)}v(x^{*})+\frac{1}{1-u(x)}v(x)$, which
implies that $v(x)=(v(x^{*})-v(y^{*}))u(x)+v(y^{*})$. Therefore, in any case,
we have that $v(x)=(v(x^{*})-v(y^{*}))u(x)+v(y^{*})$, as desired. This
completes the proof. $\blacksquare$

Lemma 2. Suppose that $\succsim$ is a weak order on $\mathscr{P}_{s}$. Then,
the following two statements are equivalent.

  1. 1)

$\succsim$ is independent and segmentally continuous.

  2. 2)

There exists an expected utility function $u:X\to\mathbb{R}$ of $\succsim$.

Because Lemma 2 is well known, we omit the proof.333This was proved by von
Neumann and Morgenstern (1944). For a modern proof, see Kreps (1988).

Suppose that $X$ is a Hausdorff topological space, and $Z=C_{b}(X)$ be the set
of all bounded and continuous real-valued functions on $X$. Let
$T^{*}(P)(u)=E_{P}[u]$. Then, $T^{*}$ is a mapping from $\mathscr{P}_{X}$ into
$Z^{\prime}$. Using this mapping, we can prove the following lemma.

Lemma 3. Suppose that $X$ is a topological space. Then, there exists a
topology on $\mathscr{P}_{X}$ such that for any net $(P_{\nu})$, this
converges to $P$ if and only if for any bounded and continuous function
$u:X\to\mathbb{R}$, the net $(E_{P_{\nu}}[u])$ converges to $E_{P}[u]$.444If
$X$ is a normal Hausdorff space, then $Z^{\prime}$ is isomorphic to the space
of all regular finitely additive Borel measures, and thus this topology is
Hausdorff. See, for example, Dunford and Schwartz (1988). However, if $X$ is
not normal, then this topology may be not Hausdorff. In particular, if $X$ is
not Tychonoff, then there may be two probability measures $P,Q$ such that
$E_{P}[u]=E_{Q}[u]$ for any bounded and measurable function $u$, and in this
case, this topology must not be Hausdorff.

Proof. As we stated, let $Z=C_{b}(X)$ be the space of all bounded and
continuous real-valued functions on $X$. For each $u\in Z$, define
$L_{u}:Z^{\prime}\to\mathbb{R}$ as $L_{u}(T)=T(u)$. Then, $L_{u}\in
Z^{\prime\prime}$. Let $\sigma_{1}$ be the set of all $L_{u}^{-1}(A)$, where
$u\in Z$ and $A$ is an open set of $\mathbb{R}$. Let $\sigma_{2}$ be the set
of all intersections of finite families in $\sigma_{1}$. Let $\tau$ be the set
of all unions of families in $\sigma_{2}$. Then, $\emptyset,X\in\sigma_{1}$,
and thus $\emptyset,X\in\tau$. If $(U_{i})$ is a family of sets in $\tau$,
then for each $i$, there exists a family $(V_{ij})$ of sets in $\sigma_{2}$
such that $U_{i}=\cup_{j}V_{ij}$. Thus,
$\cup_{i}U_{i}=\cup_{i,j}V_{ij}\in\tau$. Finally, if $U_{1},U_{2}\in\tau$,
then for $i\in\\{1,2\\}$, there exists a family $(V_{ij})$ of sets in
$\sigma_{2}$ such that $U_{i}=\cup_{j}V_{ij}$. Then, $V_{1j}\cap
V_{2j^{\prime}}\in\sigma_{2}$ for each $j,j^{\prime}$. Therefore,

$\displaystyle U_{1}\cap U_{2}=$
$\displaystyle~{}\left(\cup_{j}V_{1j}\right)\cap\left(\cup_{j}V_{2j^{\prime}}\right)$
$\displaystyle=$ $\displaystyle~{}\cup_{j,j^{\prime}}(V_{1j}\cap
V_{2j^{\prime}})\in\tau$

which implies that $\tau$ is a topology on $Z^{\prime}$.

Now, suppose that $(T_{\nu})$ is a net of $Z^{\prime}$, and $T\in Z^{\prime}$.
Suppose that $(T_{\nu})$ converges to $T$. Choose any $u\in Z$, and consider a
net $(T_{\nu}(u))$. This net is rewritten as $(L_{u}(T_{\nu}))$. Choose any
open set $A\subset\mathbb{R}$ such that $T(u)\in A$. Then, $L_{u}(T)\in A$.
Let $B=L_{u}^{-1}(A)$. Then, $B\in\tau$, and thus there exists $\nu^{*}$ such
that if $\nu\geq\nu^{*}$, then $T_{\nu}\in B$. This implies that if
$\nu\geq\nu^{*}$, then $T_{\nu}(u)\in A$. Therefore, $(T_{\nu}(u))$ converges
to $T(u)$. Conversely, suppose that for each $u\in Z$, $(T_{\nu}(u))$
converges to $T(u)$. Choose any $U\in\tau$ such that $T\in U$. By the
construction of $\tau$, there exists $V\in\sigma_{2}$ such that $T\in V$ and
$V\subset U$. Because of the definition of $\sigma_{2}$, there exist
$u_{1},...,u_{k}\in Z$ and open sets $A_{1},...,A_{k}\subset\mathbb{R}$ such
that $V=\cap_{i=1}^{k}L_{u_{i}}^{-1}(A_{i})$. For each $i$, because
$(L_{u_{i}}(T_{\nu}))$ converges to $L_{u_{i}}(T)$, there exists $\nu_{i}$
such that if $\nu\geq\nu_{i}$, $T_{\nu}(u_{i})\in A_{i}$. Because of the
definition of the direction set, there exists $\nu^{*}$ such that
$\nu^{*}\geq\nu_{i}$ for all $i\in\\{1,...,k\\}$. If $\nu\geq\nu^{*}$, then
$T_{\nu}\in V\subset U$, which implies that $(T_{\nu})$ converges to $T$.

Let $\tau^{\prime}$ be the set of all $(T^{*})^{-1}(U)$, where $U\in\tau$ and
$T^{*}:\mathscr{P}_{X}\to Z^{\prime}$ is defined as above. Then,
$\tau^{\prime}$ is a topology of $\mathscr{P}_{X}$. By definition, $(P_{\nu})$
converges to $P$ in $\tau^{\prime}$ if and only if $(T^{*}(P_{\nu}))$
converges to $T^{*}(P)$. As we proved above, $(P_{\nu})$ converges to $P$ if
and only if for each $u\in Z$, $(E_{P_{\nu}}[u])$ converges to $E_{P}[u]$.
This completes the proof. $\blacksquare$

We call this topology the weak* topology. If
$\mathscr{P}\subset\mathscr{P}_{X}$, then we also define the weak* topology on
$\mathscr{P}$ as the relative topology on $\mathscr{P}$ of the original weak*
topology. Suppose that $\succsim$ is a weak order on $\mathscr{P}$. Then, we
say that $\succsim$ is sequentially continuous if there exists a sequence
$(X_{n})$ of subsets of $X$ such that, 1) $X_{n}$ is closed, 2) $X_{n}$ is
included in the interior of $X_{n+1}$, 3) $\cup_{n}X_{n}=X$, and 4)
$\succsim_{X_{n}}$ is closed in the weak* topology.

Note that, if $X_{n}$ is clopen, then $X_{n}=X_{n+1}$ is not prohibited. In
particular, because $X$ is clopen, if $\succsim$ is closed in the weak*
topology, then it is automatically sequentially continuous.

We need the following lemma.

Lemma 4. Suppose that $\mathscr{P}$ is a convex subset of $\mathscr{P}_{c}$
that includes $\mathscr{P}_{s}$, and $\succsim$ is a weak order on
$\mathscr{P}$. If $\succsim$ is sequentially continuous, then it is
segmentally continuous.

Proof. Choose any $P,Q,R\in\mathscr{P}$ such that $P\succ Q\succ R$. Because
$\mathscr{P}\subset\mathscr{P}_{c}$, there exists a compact set $C\subset X$
such that the supports of $P,Q,R$ are included in $C$. Choose $(X_{n})$ as in
the definition of sequential continuity, and let $Y_{n}$ be the interior of
$X_{n}$. Then, $\cup_{n}Y_{n}=X$, and thus $(Y_{n})$ is an open covering of
$C$, which implies that there exists $n$ such that $C\subset Y_{n}$. Then,
$P\succ_{X_{n}}Q\succ_{X_{n}}R$, and $\succsim_{X_{n}}$ is closed with respect
to the weak* topology. Because the function $t\mapsto(1-t)P+tR$ is continuous
with respect to this topology, we have that the sets

$\\{t\in[0,1]|(1-t)P+tR\succsim_{X_{n}}Q\\},\
\\{t\in[0,1]|Q\succsim_{X_{n}}(1-t)P+tR\\}$

are closed, which is equivalent to the closedness of

$\\{t\in[0,1]|(1-t)P+tR\succsim Q\\},\ \\{t\in[0,1]|Q\succsim(1-t)P+tR\\}.$

Therefore, $\succsim$ is segmentally continuous. This completes the proof.
$\blacksquare$

##  3 Main Result

Our first main result is as follows.

Theorem 1. Suppose that $X$ is a Hausdorff topological space, and $\succsim$
is a weak order on $\mathscr{P}_{s}$. Then, there exists a continuous expected
utility function $u:X\to\mathbb{R}$ of $\succsim$ if and only if $\succsim$ is
independent and sequentially continuous. Moreover, such an expected utility
function is unique up to a positive affine transformation.

To show this theorem, we need the following lemma.

Lemma 5. Suppose that $X$ is a Hausdorff topological space, and $\succsim$ is
a weak order on $\mathscr{P}_{s}$. Then, there exists a bounded and continuous
expected utility function $u:X\to\mathbb{R}$ of $\succsim$ if and only if
$\succsim$ is independent and closed in the weak* topology.

Proof. Suppose that $u:X\to\mathbb{R}$ is a bounded and continuous expected
utility function of $\succsim$. By Lemma 1, $\succsim$ is independent. Define
$v(P,Q)=E_{P}[u]-E_{Q}[u]$. Because $u$ is bounded and continuous, $v$ is
continuous in the weak* topology. Since $\succsim=v^{-1}([0,+\infty[)$, we
have that $\succsim$ is closed, as desired.

Conversely, suppose that $\succsim$ is independent and closed. As we have
already noted, any closed weak order is sequentially continuous, and thus
$\succsim$ is also sequentially continuous. By Lemma 4, $\succsim$ is
segmentally continuous, and by Lemma 2, there exists an expected utility
function $u:X\to\mathbb{R}$ of $\succsim$.

Hereafter, $\delta_{x}$ denotes the Dirac measure whose support is $\\{x\\}$.
Suppose that $u$ is not upper semi-continuous at $x$. Then, there exists
$\varepsilon>0$ such that for any open neighborhood $U$ of $x$, there exists
$x_{U}\in U$ such that $u(x_{U})>u(x)+\varepsilon$. Clearly $(x_{U})$ is a net
that converges to $x$. Therefore, $(\delta_{x_{U}})$ converges to $\delta_{x}$
with respect to the weak* topology. Choose any open neighborhood $V^{*}$ of
$x$, and define $P=(1-t)\delta_{x}+t\delta_{x_{V^{*}}}$, where $t>0$ is
sufficiently small that $u(x)<E_{P}[u]<u(x)+\varepsilon$. Then,
$\delta_{x_{U}}\succ P$ for all $U$, but $P\succ\delta_{x}$, which contradicts
the closedness of $\succsim$. Therefore, $u$ is upper semi-continuous. By the
symmetrical arguments, we can prove that $u$ is lower semi-continuous, and
thus $u$ is continuous.

Suppose that $u$ is not bounded from above. Then, there exists $x^{*},x_{0}\in
X$ and a sequence $(x_{n})$ such that $u(x_{0})>u(x^{*})$ and
$u(x_{n})>u(x^{*})+2^{n}(u(x_{0})-u(x^{*}))$. Define
$P_{n}=(1-2^{-n})\delta_{x^{*}}+2^{-n}\delta_{x_{n}}$. Then,

$E_{P_{n}}[u]=(1-2^{-n})u(x^{*})+2^{-n}u(x_{n})>u(x_{0})=E_{\delta_{x_{0}}}[u],$

which implies that $P_{n}\succ\delta_{x_{0}}$ for all $n$. Choose any bounded
and continuous function $v:X\to\mathbb{R}$. Then,

$\lim_{n\to\infty}E_{P_{n}}[v]=\lim_{n\to\infty}[(1-2^{-n})v(x^{*})+2^{-n}v(x_{n})]=v(x^{*})=E_{\delta_{x^{*}}}[v],$

and thus, $(P_{n})$ converges to $\delta_{x^{*}}$ with respect to the weak*
topology. However, $\delta_{x_{0}}\succ\delta_{x^{*}}$, which contradicts the
closedness of $\succsim$. Therefore, $u$ is bounded from above. By the
symmetrical arguments, we can prove that $u$ is bounded from below, and thus
bounded. This completes the proof. $\blacksquare$

Proof of Theorem 1. Suppose that $u$ is a continuous expected utility function
of $\succsim$. By Lemma 1, $\succsim$ is independent. Define

$X_{n}=u^{-1}([-n,n]),Y_{n}=u^{-1}(]-n-1/2,n+1/2[).$

Then, $X_{n}\subset Y_{n}\subset X_{n+1}$ and $X=\cup_{n}X_{n}$. Because $u$
is continuous, $X_{n}$ is closed and $Y_{n}$ is open, which implies that
$X_{n}$ is included in the interior of $X_{n+1}$. Let $u_{n}$ be the
restriction of $u$ to $X_{n}$. Because $u_{n}$ is a bounded and continuous
expected utility function of $\succsim_{X_{n}}$, by Lemma 5, we have that
$\succsim_{X_{n}}$ is closed.555Note that, because $X_{n}$ is closed,
$\mathscr{P}_{X_{n}}$ can naturally be identified with the set of all regular
Borel probability measures on $X_{n}$, and thus our lemmas are applicable.
Hence, $\succsim$ is sequentially continuous.

Conversely, suppose that $\succsim$ is independent and sequentially
continuous. Then, there exists a sequence $(X_{n})$ of subsets of $X$ such
that $X_{n}$ is closed, $X_{n}$ is included in the interior of $X_{n+1}$,
$\cup_{n}X_{n}=X$, and $\succsim_{X_{n}}$ is closed with respect to the weak*
topology. Let $Y_{n}$ be the interior of $X_{n}$.

By Lemma 4, $\succsim$ is segmentally continuous, and by Lemma 2, there exists
an expected utility function $u$ of $\succsim$. Let $u_{n}$ be the restriction
of $u$ to $X_{n}$. Then, $u_{n}$ is an expected utility function of
$\succsim_{X_{n}}$.

Choose any $x\in X$. Then, there exists $n$ such that $x\in X_{n-1}$. Because
$\succsim_{X_{n}}$ is closed and independent, by Lemma 5, there exists a
bounded and continuous expected utility function $v_{n}$ of
$\succsim_{X_{n}}$. By Lemma 1, $u_{n}$ is a positive affine transformation of
$v_{n}$, and thus $u_{n}$ is bounded and continuous. Because the interior of
$X_{n}$ contains $X_{n-1}$, it is an open neighborhood of $x$, and thus $u$ is
continuous at $x$. Because $x$ is arbitrary, we conclude that $u$ is
continuous. This completes the proof. $\blacksquare$

If $X$ is a separable metric space, then this result can be extended to
$\mathscr{P}_{c}$.

Theorem 2. Suppose that $X$ is a separable metric space, and $\succsim$ is a
weak order on $\mathscr{P}_{c}$. Then, there exists a continuous expected
utility function $u$ of $\succsim$ if and only if $\succsim$ is independent
and sequentially continuous.

To prove this, we need an additional Lemma.

Lemma 6. Suppose that $X$ is a separable metric space, and $\succsim$ is a
weak order on $\mathscr{P}$, where $\mathscr{P}$ is a convex subset of
$\mathscr{P}_{X}$ that includes $\mathscr{P}_{s}$. Then, there exists a
bounded and continuous expected utility function of $\succsim$ if and only if
$\succsim$ is independent and closed in the weak* topology.

Proof. Suppose that $u:X\to\mathbb{R}$ is a bounded and continuous expected
utility function. Then, by the same arguments as in the proof of Lemma 5, we
can show that $\succsim$ is independent and closed.

Conversely, suppose that $\succsim$ is independent and closed. Let
$\succsim^{\prime}=\succsim\cap(\mathscr{P}_{s})^{2}$. Then,
$\succsim^{\prime}$ is an independent and closed weak order on
$\mathscr{P}_{s}$, and by Lemma 5, there exists a bounded and continuous
expected utility function $u:X\to\mathbb{R}$ of $\succsim^{\prime}$.

Note that, $\mathscr{P}_{s}$ is dense in $\mathscr{P}_{X}$ with respect to the
weak* topology.666See Theorem II.6.3 of Parthasarathy (1967). Choose any
$P,Q\in\mathscr{P}$. Because $u$ is continuous and bounded,
$E_{P}[u],E_{Q}[u]$ are well-defined. If $P\succ Q$, then there exist a
sequence $((P_{n},Q_{n}))$ on $\mathscr{P}_{s}^{2}$ and
$R_{1},R_{2}\in\mathscr{P}_{s}$ such that $P\succ R_{1}\succ R_{2}\succ Q$,
$P_{n}\succ R_{1}$ and $R_{2}\succ Q_{n}$ for all $n$, and
$(P_{n},Q_{n})\to(P,Q)$ as $n\to\infty$.777Choose $R_{1}$ sufficiently near to
$(2/3)P+(1/3)Q$ and $R_{2}$ sufficiently near to $(1/3)P+(2/3)Q$. Because $u$
is bounded and continuous, we have that

$\lim_{n\to\infty}E_{P_{n}}[u]=E_{P}[u],\
\lim_{n\to\infty}E_{Q_{n}}[u]=E_{Q}[u].$

Because $P_{n}\succ R_{1}\succ R_{2}\succ Q_{n}$,
$P_{n}\succ^{\prime}R_{1}\succ^{\prime}R_{2}\succ^{\prime}Q_{n}$, and thus

$E_{P_{n}}[u]>E_{R_{1}}[u]>E_{R_{2}}[u]>E_{Q_{n}}[u],$

which implies that

$E_{P}[u]\geq E_{R_{1}}[u]>E_{R_{2}}[u]\geq E_{Q}[u].$

Next, suppose that $P\succsim Q$. If there exists $R\in\mathscr{P}$ such that
$R\succ P$, then for any $t\in]0,1[$, by independence,

$tR+(1-t)P\succ tP+(1-t)P=P,$

which implies that $tR+(1-t)P\succ Q$. Therefore,

$tE_{R}[u]+(1-t)E_{P}[u]>E_{Q}[u].$

Letting $t\to 0$, we obtain $E_{P}[u]\geq E_{Q}[u]$. By the same argument, if
there exists $R\in\mathscr{P}$ such that $Q\succ R$, then we can show that
$E_{P}[u]\geq E_{Q}[u]$. Suppose that there is no $R\in\mathscr{P}_{c}$ such
that either $R\succ P$ or $Q\succ R$. If $P\sim Q$, then $P\sim R$ for all
$R\in\mathscr{P}$, which implies that $u$ is a constant function. Therefore,
$E_{P}[u]=E_{Q}[u]$. If $P\succ Q$, then we have already shown that
$E_{P}[u]>E_{Q}[u]$. Therefore, in either case, we have that $E_{P}[u]\geq
E_{Q}[u]$. Hence, $u$ is also an expected utility function of $\succsim$. This
completes the proof. $\blacksquare$

Proof of Theorem 2. Using Lemma 6 instead of Lemma 5, we can show that if
there exists a continuous expected utility function of $\succsim$, then
$\succsim$ is independent and sequentially continuous by the same
proof.888Note that, any subset of separable metric space is also a separable
metric space.

Conversely, suppose that $\succsim$ is independent and sequentially
continuous. Let $\succsim^{\prime}=\succsim\cap(\mathscr{P}_{s})^{2}$. Then,
$\succsim^{\prime}$ is an independent and sequentially continuous weak order
on $\mathscr{P}_{s}$, and by Theorem 1, there exists a continuous expected
utility function $u:X\to\mathbb{R}$ of $\succsim^{\prime}$. Choose $(X_{n})$
in the definition of sequential continuity, and let $Y_{n}$ be the interior of
$X_{n}$. Let $u_{n}$ be the restriction of $u$ to $X_{n}$. Then, $u_{n}$ is a
continuous expected utility function of $\succsim_{X_{n}}^{\prime}$.

Choose any $P,Q\in\mathscr{P}_{c}$. Then, there exists a compact set $C$ that
includes the supports of $P,Q$. Because $(Y_{n})$ is an open covering of $C$,
there exists $n$ such that $C\subset Y_{n}$. Because $\succsim_{X_{n}}$ is
independent and closed, by Lemma 6, there exists a bounded and continuous
expected utility function $v_{n}$ of $\succsim_{X_{n}}$. For any
$P^{\prime},Q^{\prime}\in\mathscr{P}_{X_{n}}\cap\mathscr{P}_{s}$,

$E_{P^{\prime}}[v_{n}]\geq E_{Q^{\prime}}[v_{n}]\Leftrightarrow
P^{\prime}\succsim_{X_{n}}Q^{\prime}\Leftrightarrow
P^{\prime}\succsim_{X_{n}}^{\prime}Q^{\prime},$

which implies that $v_{n}$ is also an expected utility function of
$\succsim_{X_{n}}^{\prime}$. By Lemma 1, $u_{n}$ is a positive affine
transformation of $v_{n}$, and thus,

$E_{P}[u]\geq E_{Q}[u]\Leftrightarrow E_{P}[u_{n}]\geq
E_{Q}[u_{n}]\Leftrightarrow E_{P}[v_{n}]\geq E_{Q}[v_{n}]\Leftrightarrow
P\succsim_{X_{n}}Q\Leftrightarrow P\succsim Q,$

which implies that $u$ is also a continuous expected utility function of
$\succsim$. This completes the proof. $\blacksquare$

##  4 Comparison with Related Results

A foundational result in this context is Lemma 2, which describes a necessary
and sufficient condition for the existence of an expected utility function.
This result was proved by von Neumann and Morgenstern (1944), and later
refined. We first mention this refinement. First, instead of considering
$\mathscr{P}_{X}$ directly, consider a structure called a mixture set. A pair
$(X,h)$ is called a mixture set if $h:[0,1]\times X^{2}\to X$ satisfies the
following three conditions. First, $h(1,x,y)=x$. Second,
$h(t,x,y)=h(1-t,y,x)$. Third, $h(t,h(s,x,y),y)=h(st,x,y)$. If there is no risk
of confusion, we use the notation $tx+(1-t)y$ instead of $h(t,x,y)$. Then, the
three requirements can be rewritten as 1) $1x+0y=x$, 2) $tx+(1-t)y=(1-t)y+tx$,
and 3) $t(sx+(1-s)y)+(1-t)y=stx+(1-st)y$, which seem to be quite natural. As
can be easily verified, any convex subset of $\mathscr{P}_{X}$ is a mixture
set. Therefore, Lemma 2 can be refined using the existence theorem for a real-
valued function that represents a weak order in the mixture set. This method
was established by Herstein and Milnor (1953). It can therefore be stated that
Neumann–Morgenstern’s result for the existence of an expected utility function
was built on the basis of the convex structure, not the topological one.

As an existence theorem for a real-valued function representing a weak order,
Neumann–Morgenstern’s theorem is the second oldest after Alt (1936). The most
prominent theorem in this context is Debreu’s (1954). We shall briefly mention
Debreu’s theorem. Let $X$ be a separable Hausdorff topological space and let
$\succsim$ be a weak order on $X$. Then, $\succsim$ is closed as a subset of
$X^{2}$ if and only if there exists a continuous representation
$u:X\to\mathbb{R}$ of $\succsim$. By comparison, Neumann–Morgenstern’s result
does not require any topological structure on $X$, and thus it is independent
of Debreu’s result. As a corollary, however, even if $X$ is a topological
space, nothing can be said about the continuity of Neumann–Morgenstern’s
expected utility function from their result alone.

In several studies, researchers have attempted to search for a natural
condition for the expected utility function to be continuous. Grandmont (1972)
is one of the most representative ones. He proved our Lemma 6 in the case
where $\mathscr{P}$ is closed. He used the approach of functional analysis for
the proof. Let us explain the main idea of his proof. First, for $Z=C_{b}(X)$
in the proof of Lemma 3, if $X$ is a separable metric space, then
$\mathscr{P}_{X}$ can naturally be seen as a subset of $Z^{\prime}$.999In
particular, if $X$ is locally compact, then by Riesz–Markov–Kakutani’s
representation theorem, $Z^{\prime}$ is isometric to the space of all Radon
measures. Grandmont considered $\mathscr{P}$ as a convex subset of
$Z^{\prime}$ and introduced the weak* topology. In this case, since
$\mathscr{P}$ is a separable metric space with respect to the usual Prohorov
metric, it follows from Debreu’s theorem that the weak order $\succsim$ on
$\mathscr{P}$ is weak* closed if and only if there exists a weak* continuous
representation function $U(P)$. He further applied Herstein–Milnor’s result
and showed that if, additionally, $\succsim$ is independent, then $U(P)$ can
be extended to a weak* continuous linear functional defined on the space of
Radon measures. It can be proved from a very elementary proposition on the
duality that for such $U$, there must exist a function $u\in Z$ such that
$U(P)=E_{P}[u]$, and thus Lemma 6 has now been proved.

However, the fact that $u\in Z$ means that $u$ must be bounded. A typical
application of the theory of expected utility is the case $X=]0,+\infty[$ and
$u(x)=\frac{x^{1-\theta}-1}{1-\theta}$. ($\theta>0$ is always assumed, and if
$\theta=1$, we consider $u(x)=\log x$.) In this case, $u$ is unbounded for any
$\theta>0$, and therefore Grandmont’s result is not applicable to this model.
In other words, the assumption that $\succsim$ is weak* closed is too strong
for standard applications.

This paper attempted to solve this problem by introducing a new condition
named sequential continuity. This approach has the advantage that if
$\mathscr{P}=\mathscr{P}_{s}$, then a very weak topological assumption on $X$
is needed. In fact, the only assumption we require in Theorem 1 is that $X$ is
a Hausdorff topological space. The reason why the Hausdorff property is
necessary is that otherwise an element of $\mathscr{P}_{s}$ may not be a Borel
measure. However, this assumption is extremely weak, and thus it cannot even
be proved that $\mathscr{P}_{s}$ is Hausdorff. If $X$ is normal, then the
Hausdorff property of $\mathscr{P}_{s}$ can be easily proved using Urysohn’s
lemma. In fact, a weaker assumption is sufficient, i.e., if $X$ is Tychonoff,
then $\mathscr{P}_{s}$ is Hausdorff. However, if $X$ is a topological space
that is not even Tychonoff, then there is a possibility of the existence of
two probabilities $P,Q\in\mathscr{P}_{s}$ such that $E_{P}[v]=E_{Q}[v]$ for
any bounded continuous function $v:X\to\mathbb{R}$. If so, any open set
containing $P$ also contains $Q$, and thus $\mathscr{P}_{s}$ is not Hausdorff.
We have carefully check the seemingly well-known property of weak* topology in
Lemma 3 in order to carefully verify that our proof is valid even in such an
odd topological space.

Unfortunately, our result can only be discussed up to the case
$\mathscr{P}\subset\mathscr{P}_{c}$. If $P$ were not compact-support, then
$E_{P}[u]$ may not be definable for unbounded $u$. Since our aim is to derive
a necessary and sufficient condition for the expected utility function to be
continuous, we have to accept this restriction as unavoidable. If some growth
condition can be introduced in addition to continuity, then there are several
known representation theorems for a continuous expected utility function on
spaces including some $P$ that is not compact-support. See, for example,
Dillenberger and Krishna (2014).

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