Teoriya kis rozdil topologiyi ta algebri vivchaye kosi i grupi kis skladeni z yihnih klasiv ekvivalentnosti Priklad kosi z troma dugami Zmist 1 Viznachennya kosi 2 Grupa kis 3 Spletennya 4 Div takozh 5 Primitki 6 Literatura 7 PosilannyaViznachennya kosi RedaguvatiKosa z n displaystyle n nbsp nitok ob yekt sho skladayetsya z dvoh paralelnih ploshin P 0 displaystyle P 0 nbsp i P 1 displaystyle P 1 nbsp u trivimirnomu prostori R 3 displaystyle mathbb R 3 nbsp yaki mistyat uporyadkovani mnozhini tochok a 1 a 2 a n P 0 displaystyle a 1 a 2 dots a n in P 0 nbsp i b 1 b 2 b n P 1 displaystyle b 1 b 2 dots b n in P 1 nbsp i z n displaystyle n nbsp roz yednanih mizh soboyu prostih dug l 1 l 2 l n displaystyle l 1 l 2 dots l n nbsp yaki peretinayut kozhnu paralelnu ploshinu P t displaystyle P t nbsp mizh P 0 displaystyle P 0 nbsp i P 1 displaystyle P 1 nbsp odnorazovo i z yednuyut tochki a i displaystyle a i nbsp z tochkami b i displaystyle b i nbsp Zazvichaj vvazhayetsya sho tochki a 1 a 2 a n displaystyle a 1 a 2 dots a n nbsp lezhat na pryamij l 0 displaystyle l 0 nbsp v P 0 displaystyle P 0 nbsp a tochki b 1 b 2 b n displaystyle b 1 b 2 dots b n nbsp na pryamij l 1 displaystyle l 1 nbsp v P 1 displaystyle P 1 nbsp paralelnij l 0 displaystyle l 0 nbsp prichomu a i displaystyle a i nbsp roztashovani pid b i displaystyle b i nbsp dlya kozhnogo i displaystyle i nbsp Kosi zobrazhuyutsya v proyekciyi na ploshinu sho prohodit cherez l 0 displaystyle l 0 nbsp i l 1 displaystyle l 1 nbsp cya proyekciya mozhe buti zvedena v zagalne polozhennya tak sho ye lishe skinchenne chislo podvijnih tochok poparno roztashovanih na riznih rivnyah i peretini transversalni Grupa kis RedaguvatiDokladnishe Grupa kisU mnozhini vsih kis z n nitkami i z fiksovanimi P 0 P 1 a i b i displaystyle P 0 P 1 a i b i nbsp vvoditsya vidnoshennya ekvivalentnosti Vono viznachayetsya gomeomorfizmami h P P displaystyle h Pi to Pi nbsp de P displaystyle Pi nbsp oblast mizh P 0 displaystyle P 0 nbsp i P 1 displaystyle P 1 nbsp totozhnimi na P 0 P 1 displaystyle P 0 cup P 1 nbsp Kosi a displaystyle alpha nbsp i b displaystyle beta nbsp ekvivalentni yaksho isnuye takij gomeomorfizm h displaystyle h nbsp sho h a b displaystyle h alpha beta nbsp Klasi ekvivalentnosti dali takozh zvani kosami utvoryuyut grupu kis B n displaystyle B n nbsp Odinichna kosa klas ekvivalentnosti yakij mistit kosu z n paralelnih vidrizkiv Kosa a 1 displaystyle alpha 1 nbsp zvorotna do kosi a displaystyle alpha nbsp viznachayetsya vidobrazhennyam u ploshini P 1 2 displaystyle P 1 2 nbsp Nitka kosi z yednuye a i displaystyle a i nbsp z b j i displaystyle b j i nbsp i viznachaye pidstanovku element simetrichnoyi grupi S n displaystyle S n nbsp Yaksho cya pidstanovka totozhna to kosa nazivayetsya farbovanoyu abo chistoyu kosoyu Ce vidobrazhennya zadaye epimorfizm B n displaystyle B n nbsp na grupu S n displaystyle S n nbsp perestanovok n elementiv yadrom yakogo ye pidgrupa K n displaystyle K n nbsp yaka vidpovidaye vsim chistim kosam tak sho ye korotka tochna poslidovnist 0 K n B n S n 0 displaystyle 0 to K n to B n to S n to 0 nbsp Spletennya RedaguvatiNehaj C displaystyle mathcal C nbsp tenzorna kategoriya Spletennyam u C displaystyle mathcal C nbsp ye struktura komutuvannya na C displaystyle mathcal C nbsp yaka zadovolnyaye dvom spivvidnoshennyam a U V W a U W 1 V 1 V a V W displaystyle a U otimes V W a U W otimes 1 V 1 V otimes a V W nbsp a U V W 1 V a U W a U V 1 W displaystyle a U V otimes W 1 V otimes a U W a U V otimes 1 W nbsp dlya usih ob yektiv U V W displaystyle U V W nbsp Yaksho c displaystyle c nbsp spletennya u C displaystyle mathcal C nbsp to j c 1 displaystyle c 1 nbsp ye spletennyam u C displaystyle mathcal C nbsp Kosovoyu monoyidalnoyu kategoriyeyu ye monoyidalna kategoriya osnashena spletennyam Nehaj V displaystyle V nbsp vektornij prostir nad k displaystyle k nbsp Rivnyannya Yanga Bakstera rivnyannya dlya linijnogo avtomorfizmu z prostoru V V displaystyle V otimes V nbsp a 1 V 1 V a a 1 V 1 V a a 1 V 1 V a displaystyle a otimes 1 V 1 V otimes a a otimes 1 V 1 V otimes a a otimes 1 V 1 V otimes a nbsp Ce rivnyannya ye rivnistyu elementiv grupi avtomorfizmiv V V V displaystyle V otimes V otimes V nbsp Jogo rozv yazok nazivayetsya R displaystyle R nbsp matriceyu Dlya vektornogo prostoru V displaystyle V nbsp cherez t V V A u t V V displaystyle tau V V in mathrm Aut V otimes V nbsp poznachimo operator perestanovki spivmnozhnikiv yakij predstavlyaye dvi kopiyi cogo prostoru Vin viznachayetsya spivvidnoshennyamt V V v 1 v 2 v 2 v 1 v 1 v 2 V displaystyle tau V V v 1 otimes v 2 v 2 otimes v 1 quad forall v 1 v 2 in V nbsp Operator perestanovki zadovolnyaye rivnyannyu Yanga Bakstera oskilki v simetrichnij grupi vikonuyetsya spivvidnoshennya Kokstera 1 R 12 R 23 R 12 R 23 R 12 R 23 displaystyle R 12 R 23 R 12 R 23 R 12 R 23 nbsp de verhni indeksi i j displaystyle ij nbsp viznachayut transpoziciyu yaka minyaye i displaystyle i nbsp ta j displaystyle j nbsp Nehaj A A 0 A 1 displaystyle A A 0 oplus A 1 nbsp asociativna algebra iz odiniceyu nad deyakim algebrichno zamknenim polem nulovoyi harakteristiki k displaystyle mathbb k nbsp na yakij viznachena operaciya kodobutku D displaystyle Delta nbsp zadani antipod S displaystyle S nbsp ta kosij antipod S displaystyle S nbsp tobto antipod dlya protilezhnogo kodobutku D o p displaystyle Delta op nbsp a takozh odinicya e displaystyle e nbsp S 1 S displaystyle S 1 S nbsp A R displaystyle A R nbsp ye kvazitrikutnoyu superalgebroyu Hopfa yaksho R displaystyle R nbsp zadovolnyaye kvantovomu rivnyannyu Yanga Bakstera R 12 R 13 R 23 R 23 R 13 R 12 displaystyle R 12 R 13 R 23 R 23 R 13 R 12 nbsp a takozh spivvidnoshennyam 2 S 1 R 1 S 1 R R 1 displaystyle S otimes 1 R 1 otimes S 1 R R 1 nbsp e 1 R 1 e R 1 displaystyle e otimes 1 R 1 otimes e R 1 nbsp S 1 R 1 1 S 1 R 1 R displaystyle S otimes 1 R 1 1 otimes S 1 R 1 R nbsp Div takozh RedaguvatiTeoriya vuzliv Grupa kisPrimitki Redaguvati Christian Kassel Marc Rosso Vladimir Turaev Quantum and knot invariants Stukopin Vladimir Alekseevich Angiany superalgebry Li Literatura RedaguvatiSosinskij A Kosy i uzly nedostupne posilannya z veresnya 2019 Kvant 2 1989 stor 6 14 ros Posilannya RedaguvatiTanec pro teoriyu kis navryad chi vi koli nebut bachili shos podibne 2017r Otrimano z https uk wikipedia org w index php title Teoriya kis amp oldid 39982218
