PPT

... 1) u(L) = 0 for any unlink L; 2) u(L) = min u(L-)+1, where the minimum is taken over all minimal projections of links L-, obtained from a minimal projection of L by one crossing change. This means that we take a minimal projection of a link L, make a crossing change in every crossing, and minimize a ...

... 1) u(L) = 0 for any unlink L; 2) u(L) = min u(L-)+1, where the minimum is taken over all minimal projections of links L-, obtained from a minimal projection of L by one crossing change. This means that we take a minimal projection of a link L, make a crossing change in every crossing, and minimize a ...

by Marta Kobiela

... We will always be able to do this for one of two reasons. In the first one, open vertices already exist to make this addition possible. If these vertices do not exist, it means that those vertices must be occupied by another stick. Repeating this process, we eventually get a stick at the very edge o ...

... We will always be able to do this for one of two reasons. In the first one, open vertices already exist to make this addition possible. If these vertices do not exist, it means that those vertices must be occupied by another stick. Repeating this process, we eventually get a stick at the very edge o ...

Knot energies and knot invariants

... etc.. And there is always the moduli problem if geometric objects with maximal homogeneity are not unique. So, we may also ask for smooth imbeddings S 1 → R3 , which we will refer to as geometric knots, with the “most perfect” shape among all geometric knots isotopic to each other. This geometric si ...

... etc.. And there is always the moduli problem if geometric objects with maximal homogeneity are not unique. So, we may also ask for smooth imbeddings S 1 → R3 , which we will refer to as geometric knots, with the “most perfect” shape among all geometric knots isotopic to each other. This geometric si ...

Full text

... representations are not in general unique (that is* a given knot may have more than one representation), but it is conjectured that any representation is an invariant of that knot* For example* the knot (32) shown below may be represented in the following ways, by various vertex deletions: ...

... representations are not in general unique (that is* a given knot may have more than one representation), but it is conjectured that any representation is an invariant of that knot* For example* the knot (32) shown below may be represented in the following ways, by various vertex deletions: ...

Quipus, sometimes known as khipus or talking knots, were recording devices historically used in the region of Andean South America. A quipu usually consisted of colored, spun, and plied thread or strings made from cotton or camelid fiber. For the Inca, the system aided in collecting data and keeping records, ranging from monitoring tax obligations, properly collecting census records, calendrical information, and military organization. The cords contained numeric and other values encoded by knots in a base ten positional system. A quipu could have only a few or up to 2,000 cords. The configuration of the quipus have also been ""compared to string mops."" Archaeological evidence has also shown a use of finely carved wood as a supplemental, and perhaps more sturdy, base on which the color-coordinated cords would be attached. A relatively small number have survived.Objects that can be identified unambiguously as quipus first appear in the archaeological record in the first millennium CE. They subsequently played a key part in the administration of Tahuantinsuyu, the empire controlled by the Incan ethnic group, which flourished across the Andes from c. 1450 to 1532 CE. As the region was subsumed under the invading Spanish Empire, the use of the quipu faded from use, to be replaced by European writing systems. However, in several villages, quipu continued to be important items for the local community, albeit for ritual rather than recording use. It is unclear as to where and how many intact quipus still exist, as many have been stored away in mausoleums, 'along with the dead.'Quipu is the Spanish spelling and the most common spelling in English. Khipu (pronounced [ˈkʰipu]) is the word for ""knot"" in Cusco Quechua (the native Inca language); the kh is an aspirated k. In most Quechua varieties, the term is kipu.