In the branch of mathematics called knot theory, the volume conjecture is the following open problem that relates quantum invariants of knots to the hyperbolic geometry of knot complements. Let O denote the unknot. For any knot K let ⟨ K ⟩ N be Kashaev's invariant of K. Rinat Kashaev observed that the asymptotic behavior of a certain state sum of knots gives the hyperbolic volume vol of the complement of knots K and showed that it is true for the knots 4 1, 5 2, 6 1, he conjectured. His invariant for a knot K is based on the theory of quantum dilogarithms at the N -th root of unity, q = exp . Murakami & Murakami had firstly pointed out that Kashaev's invariant is related to Jones polynomial by replacing q with the 2N-root of unity, exp i π N, they used an R-matrix as the discrete Fourier transform for the equivalence of these two values. The volume conjecture is important for knot theory. In section 5 of this paper they state that: Assuming the volume conjecture, every knot, different from the trivial knot has at least one different Vassiliev invariant.
Using complexification, Murakami et al. rewrote the formula into where C S is called the Chern–Simons invariant. They showed that there is a clear relation between the complexified colored Jones polynomial and Chern–Simons theory from a mathematical point of view. Kashaev, Rinat M. "The hyperbolic volume of knots from the quantum dilogarithm", Letters in Mathematical Physics, 39: 269–275, arXiv:q-alg/9601025, doi:10.1023/A:1007364912784. Murakami, Hitoshi. Murakami, Hitoshi. Gukov, Sergei, "Three-Dimensional Quantum Gravity, Chern-Simons Theory, And The A-Polynomial ", Commun. Math. Phys. 255: 557–629, arXiv:hep-th/0306165, Bibcode:2005CMaPh.255..577G, doi:10.1007/s00220-005-1312-y
Erotic spanking is the act of spanking another person for the sexual arousal or gratification of either or both parties. It is considered part of BDSM, it may involve light and brief spanking or much more extensive spanking, including the use of implements such as whips or paddles. Activities range from a spontaneous smack on bare buttocks during a sexual activity, to occasional sexual roleplay, such as ageplay, to domestic discipline and may involve the use of a hand or the use of a variety of spanking implements, such as a spanking paddle or cane. Erotic spankings are combined with other forms of sexual foreplay; the most common type of erotic spanking is administered on the bare buttocks. For safety, erotic spanking should take care to avoid hipbones. Spanking can be combined with bondage, in order to heighten the sexual stimulation of the experience. See also: Spanking literature One of the earliest depictions of erotic spanking is found in the Etruscan Tomb of the Whipping from the fifth century BC.
Spanking was a constant focus of erotica produced during the Victorian era. Interest in sexual gratification received from giving and witnessing spanking began to increase during the 19th century. Interest was not only confined to spanking literature, but the development of photography during 19th century resulted in the beginning of the creation of spanking photography. In the context of literature, this growing demand caused the publication of numerous limited edition spanking novels; this interest for spanking followed into the next century, with the early 20th century being considered the "Golden Age" of spanking literature. This period of spanking literature is marked by three notable characteristics. First, greater audiences were reached with the availability of less expensive editions and greater print runs. Second, many of the spanking novels contained numerous illustrations. Third, this period saw a gradual increase in the output and publication of spanking literature, growing within the 1920s and peaking within the 1930s.
Much of the output of spanking literature during this period was by French publishers and illustrators. Within the context of spanking photography, France was the home to the creation of much content, with the most notable studios being the Biederer Studio and the Ostra Studio; this "Golden Age" of spanking literature came to an end as a result of the Second World War, more due to the German occupation of France between 1940 and 1944 and the enforcement of censorship laws. A somewhat notable exception to the decline of spanking literature during this period was John Willie's bondage Bizarre magazine. Of the many French works from the "Golden Age" few at the time were translated into other languages within which spanking literature was popular, namely English and German, but beginning during the mid-1960s a number of these French works were translated into English and published, along with these works being republished in French and older British works being republished; the occurrence of this was facilitated by the availability of mass-produced paperbacks and changes in censorship laws.
According to Dan Savage, journalist Jillian Keenan is "America's most prominent spanking fetishist" today. She has written about erotic spanking for the New York Times and Pacific Standard. In her Slate article, Keenan criticized the use of spanking as a punishment for children, while acknowledging that children and adolescents who experience sexual pleasure while being beaten may go on to enjoy the fetish of sadomasochism as adults. Keenan has argued that spanking fetishism is a form of sexual orientation, which should not be considered a mental illness. Routine corporal punishment of women by their husbands, including spanking, does still exist in some parts of the world. However, spanking of an adult tends to be confined to erotic spanking. A controversial subculture known as Christian domestic discipline promotes spanking of wives by their husbands as a form of punishment. Christian conservative radio host Bryan Fischer said to the Huffington Post that it was a "horrifying trend – bizarre, unbiblical and un-Christian".
A spanking may be carried out with the use of a bare or gloved hand, or with any of a variety of implements, including a paddle, strap, feather duster or belt. Other popular implements are canes, riding crops, switches, sneakers, rolled-up catalogs or newspapers, rulers or martinets. A spank skirt or spanking skirt has an additional opening in back designed to expose the buttocks. While the name spank skirt suggests that the wearer could be spanked "bare bottom" without removing or repositioning the skirt, this item may be worn for reasons other than spanking, for instance, exposure. Considered fetish wear, spank skirts are tight-fitting and made of fetishistic materials. Regardless of the gender of the wearer, spank skirts are considered female attire; the male gender role equivalent might be motorcycle chaps. A spanking bench or spanking horse is a piece of furniture used to position a spankee on, with or without restraints, they come in many sizes and styles, the most popular of, similar to a sawhorse with a padded top and rings for