Knots and colours
Ukurasa huu bado haujatafsiriwa kwa Kiswahili, kwa hiyo unaonyeshwa kwa English.
Saidia kutafsiri →Swali
Tie a knot in a piece of string and glue the two ends together. How could you prove that it really is knotted — that no amount of wiggling, without cutting, will turn it into a plain loop?
Fanya jaribio kabla ya kuendelea kusoma. Hakuna njia mbaya ya kuchunguza.
Jaribio
Kupaka mafundo rangi tatu
Paka rangi kila tao la mchoro. Kwenye kila kivuko, matao matatu yanayokutana lazima yawe ya rangi moja yote au ya rangi tofauti zote.
Bofya tao ili kubadilisha rangi yake (au lichague kisha ubonyeze Enter, au 1, 2, 3). Tao huanzia mahali uzi unapopita chini hadi mahali unapopita chini tena.
- Vivuko katika mchoro huu
- 3
- Namba ya vivuko (idadi ndogo kabisa)
- 3
Matao yaliyopakwa rangi: 0 kati ya 3.
Kwa nini kanuni hii? Michoro yoyote miwili ya fundo moja huhusiana kwa aina tatu za miondoko midogo, miondoko ya Reidemeister: kuongeza au kuondoa msokoto, kutelezesha uzi mmoja juu ya mwingine, na kutelezesha uzi kupita kivuko. Kila mwondoko hugeuza upakaji rangi halali wenye rangi mbili au zaidi kuwa upakaji mwingine halali.
Fundo la majani matatu linaweza kupakwa rangi tatu na fundo tupu haliwezi. Kwa kuwa uwezo huo ni kisichobadilika, ulisogeze vyovyote, fundo la majani matatu halitakuwa fundo tupu: kweli limefungwa.
Kwa hiyo uwezo wa kupakwa rangi tatu ni kisichobadilika cha mafundo: kama mchoro mmoja wa fundo unaweza kupakwa rangi tatu, kila mchoro wake unaweza. Duara tupu lina tao moja tu, kwa hiyo fundo tupu haliwezi kupakwa rangi tatu.
Vidokezo, ukivitaka
Kinachoendelea
Kinachoendelea
A knot diagram is a network drawing with crossings, like the ones in the dots-and-lines lesson, except that at each crossing we record which strand goes over. Two diagrams show the same knot exactly when one can be changed into the other by moving the string around — and in 1927 Kurt Reidemeister proved that it is enough to use three simple local moves: adding or removing a twist, sliding one strand over another, and sliding a strand past a crossing.
The colouring rule in the hints is called tricolouring. Each of Reidemeister's moves can be checked to keep a good colouring good (and to keep "at least two colours" true). So whether a knot can be tricoloured does not depend on how it is drawn: it is a knot invariant.
A plain loop drawn with no crossings has a single arc, so it can only ever use one colour: it is not tricolourable. The trefoil is. Therefore the trefoil is genuinely knotted — no wiggling will undo it. That is a proof, not just evidence.
Invariants have limits. The figure-eight knot cannot be tricoloured either, yet it is knotted too; showing that needs stronger invariants, such as colourings with five colours or the Jones polynomial (Vaughan Jones, 1984). Remarkably, there are algorithms that decide whether any given diagram is the unknot (Wolfgang Haken, 1961) — but whether the Jones polynomial alone can always recognise the unknot is still an open question.
Hii inaelekea wapi
- Shahada ya kwanza (juu) · Somo linapatikanaKnot theoryKnots and links in space and their diagrams, Reidemeister moves, invariants that tell knots apart (tricolourability and Fox colourings, linking number, the Kauffman bracket and Jones polynomial), Seifert surfaces and genus, and an honest map of what is known and open.
- Njia za utafiti · Somo linapatikanaLow-dimensional topology: 3- and 4-manifoldsOptional research direction: 3-manifolds through Heegaard splittings, surgery and the eight Thurston geometries; the Poincaré conjecture and geometrisation (Perelman); hyperbolic 3-manifolds; and the strange world of 4-manifolds (Freedman, Donaldson, exotic ℝ⁴), separating established results from open questions.
- Njia za utafiti · Somo linapatikanaCategorical representation theoryOptional research direction: representations on categories rather than vector spaces, including tensor and fusion categories, module categories, categorical actions and categorification, such as Chuang–Rouquier sl₂-categorification and Khovanov homology.