Energy of knots was introduced to produce an ``optimal embedding"
of a knot, a beautiful knot which would represent its knot type. 
The basic philosophy due to Fukuhara and Sakuma independently is as follows. 

Suppose there is a non-conductive knotted string which is charged uniformly 
in a non-conductive viscous fluid. 
Then it might evolve to decrease its electrostatic energy 
 without intersecting itself because of Coulomb's repulsive force 
 until it comes to a critical point of the energy. 
%Then an optimal configuration of a knot might be  realized as the limit of this self-evolving process. 
Hence we might be able to define an ``optimal embedding" of a knot 
by an ``energy minimizer", 
which is an embedding that attains the minimum energy within its isotopy class. 
Thus our motivational problem can be stated as: 

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Can we define an ``energy" on the space of knots 
so that its gradient flow can evolve a knot 
to an ``optimal embedding" while preserving its isotopy type? 
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For this purpose, our energy should blow up if a knot 
degenerates to a ``singular knot" with double points 
since a knot might change its knot type through a crossing change. 

The first example of such an energy, $E^{(2)}$, for smooth knots 
 was defined as the renormalization of a modified 
 electrostatic energy of charged knots under the assumption that 
 the absolute value of the repulsive force between a pair of unit point charges 
 is proportional to the inverse cube of the distance. 

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In Part 1 we introduce several types of the knot energy and study 
the problem of whether or not there is an ``energy minimizer" in each knot type. 
We first give a family of repulsive energies, 
which are obtained as the renormalization 
of modified electrostatic energy of charged knots. 
Then the answer to the above problem may depend on the following three conditions;  
the power exponent with which the distance between a pair of points on a knot
are integrated, 
the primeness of a knot, which is a topological condition, and 
the metric of the ambient space, which is a geometric condition. 
We then introduce geometrically defined knot energies of another kind, 
such as ropelength. 
Roughly speaking they measure how much a knot can be fattened, 
or how tight a rope can be knotted. 

The interest in knot energies arose naturally 
when we regarded the knots as real objects. 
They have concrete backgrounds and applications 
in various fields of natural science. 
We introduce some results of numerical experiments. 

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In Part 2 we study conformal geometry. 
The results in Part 2 were obtained in joint work with R\'emi Langevin, 
summarized in \cite{La-OH1}, from which we cite frequently. 
The first example of the knot energy, $E^{(2)}$, was eventually proved to be 
invariant under M\"obius transformations of $\mathbb{R}^3\cup\{\infty\}$ by 
Freedman, He, and Wang. 
Allowing the action of a large group as the M\"obius group 
has both advantages and disadvantages. 
If a knot is an energy minimizer then its image by 
any M\"obius transformation is again an energy minimizer 
of the same knot type (or possibly, its mirror image). 
On the other hand, conformal geometry could provide a new viewpoint 
to knot theory. 
We use the set of spheres in $S^3$, which forms a $4$ dimensional 
 hyperbolic hypersurface in Minkowski space $\mathbb{R}^{4,1}$. 
Using a sphere that is tangent to a knot $K$ at two points 
we introduce a complex valued (meromorphic) $2$-form on $K\times K$ which will be called 
 the infinitesimal cross ratio. 
This $2$-form will give another interpretation of $E^{(2)}$ and also 
 it can express other conformally invariant energies. 
Finally an energy defined from an integral geometric viewpoint 
will be introduced. 

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The energies presented in this book are defined geometrically 
and measure the complexity of embeddings. 
The study of our these subjects is called physical knot theory. 

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{\large{\bf Acknowledgments}}

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The author is sincerely grateful to 
Jason Cantarella, Yutaka Kanda, Nariya Kawazumi, Masanori Kobayashi, Rob Kusner, Tomotada Ohtsuki, Kaoru Ono, Hiro-o Tokunaga, Andrzej Stasiak, Takashi Tsuboi, 
and especially, R\'emi Langevin, for helpful suggestions and conversations. 

The author deeply thanks his colleague Martin Guest for correcting the English translation. 

He also thanks Rob Kusner and John Sullivan for the use of their figures 
of the optimal knots in Section \ref{section_ks_pix}. 

The author appreciates the kind hospitality of 
Laboratoire de Topologie, Universit\'e de Bourgogne 
during his stay in Dijon, France to study with R\'emi Langevin. 

Some of the sections in Part 2, 
in particular Section \ref{section_jump_e_mnts}, 
are based on the manuscript for \cite{La-OH1} by Langevin.