Òåçèñû ìåæäóíàðîäíîé êîíôåðåíöèè ¾Äèíàìèêà â Ñèáèðè¿ Î ÅÂÊËÈÄÎÂÎÉ ÑÒÐÓÊÒÓÐÅ ÍÀ ÓÇËÅ ÂÎÑÜÌÅÐÊÀ Ñ ÌÎÑÒÎÌ ÄÀÐÜß ÞÐÜÅÂÍÀ ÑÎÊÎËÎÂÀ Öåëü íàñòîÿùåé ðàáîòû èçó÷åíèå åâêëèäîâûõ ñòðóêòóð íà óçëàõ è çàöåïëåíèÿõ.  1975 ãîäó Ð. Ðàéëè [1] îáíàðóæèë ïðèìåðû ãèïåðáîëè÷åñêèõ ñòðóêòóð íà íåêîòîðûõ óçëàõ è äîïîëíåíèÿõ çàöåïëåíèé â òðåõìåðíîé ñôåðå. Ïîçäíåå, âåñíîé 1977, Ó. Ï. Òåðñòîí ïðåäñòàâèë òåîðåìó ñóùåñòâîâàíèÿ äëÿ ðèìàíîâîé ìåòðèêè ïîñòîÿííîé îòðèöàòåëüíîé êðèâèçíû íà òðåõìåðíûõ ìíîãîîáðàçèÿõ. Íà ïðàêòèêå îêàçàëîñü, ÷òî äîïîëíåíèÿ ïðîñòûõ óçëîâ, èñêëþ÷àÿ òîðè÷åñêèå è ñàòòåëèòíûå, äîïóñêàþò ãèïåðáîëè÷åñêóþ ñòðóêòóðó. Îòìåòèì ñëåäóþùèé èçâåñòíûé ðåçóëüòàò: åâêëèäîâà ñòðóêòóðà íà óçëå âîñüìåðêà 41 2π . Ýòîò ðåçóëüòàò áûë ïîëó÷åí Òåðñòîâîçíèêàåò, êîãäà åãî êîíè÷åñêèé óãîë α ðàâåí 3 íîì [2]. ßâíàÿ êîíñòðóêöèÿ ôóíäàìåíòàëüíîãî ìíîæåñòâà äëÿ êîíè÷åñêîãî ìíîãîîáðàçèÿ 41 (α) â E3 áûëà ïðåäëîæåíà â ðàáîòå À. Ä. Ìåäíûõ è À. À. Ðàññêàçîâà [3]. Ýòî ôóíäàìåíòàëüíîå ìíîæåñòâî ïðåäñòàâëÿåò ñîáîé íåâûïóêëûé äâàäöàòèãðàííèê, âåðøèíû êîòîðîãî çàäàþòñÿ öåëî÷èñëåííûìè êîîðäèíàòàìè. Âîïðîñ ñóùåñòâîâàíèÿ åâêëèäîâîé ñòðóêòóðû íà çàöåïëåíèè Óàéòõåäà èçó÷åí â ðàáîòå Ð. Í. Øìàòêîâà [4].  ðàáîòå àâòîðà [5] áûëî èññëåäîâàíî ñòðîåíèå ôóíäàìåíòàëüíîãî ìíîãîãðàííèêà äëÿ óçëà òðèëèñòíèê ñ ìîñòîì è äàíû óñëîâèÿ ñóùåñòâîâàíèÿ åâêëèäîâîé ñòðóêòóðû ñîîòâåòñòâóþùåãî êîíè÷åñêîãî ìíîãîîáðàçèÿ.  äàííîé ðàáîòå ìû èññëåäóåì îñíîâíûå ãåîìåòðè÷åñêèå èíâàðèàíòû åâêëèäîâà êîíè÷åñêîãî ìíîãîîáðàçèÿ, ñèíãóëÿðíûì ìíîæåñòâîì êîòîðîãî ÿâëÿåòñÿ óçåë âîñüìåðêà ñ ìîñòîì, à íîñèòåëåì òðåõìåðíàÿ ñôåðà. Ìû óñòàíîâèì óñëîâèÿ ñóùåñòâîâàíèÿ òàêîãî îáúåêòà è âû÷èñëèì åãî îáúåì. Ñïèñîê ëèòåðàòóðû [1] Riley R. An elliptical path from parabolic representations to hyperbolic structure in Topology of Low-Dimension manifolds, LNM, 722, Springer, 99-133 (1979). [2] Thurston W. The geometry and topology of 3-manifold. Lecture Notes. Princeton University, (1980). [3] Mednykh A., Rasskazov A. Volumes and degeneration of cone-structures on the gureeight knot. Tokyo J. Math. Vol. 29, No. 2, 445-464 (2006). [4] Shmatkov R.N. Properties of Euclidean Whitehead link cone-manifolds Sib. Adv. Math., Vol. 13, No. 1. 55-86 (2003). [5] Ñîêîëîâà Ä.Þ. Î ñóùåñòâîâàíèè åâêëèäîâîé ñòðóêòóðû íà óçëå òðèëèñòíèê ñ ìîñòîì Ìàò. çàìåòêè ßÃÓ, Òîì 20, 1. 128-140 (2013). Ñîêîëîâà Äàðüÿ Þðüåâíà, Íîâîñèáèðñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, óë. Ïèðîãîâà, 2, ã. Íîâîñèáèðñê, 630090, Ðîññèÿ Èíñòèòóò ìàòåìàòèêè èì. Ñ.Ë. Ñîáîëåâà, ïð. Êîïòþãà 4, ã. Íîâîñèáèðñê, 630090, Ðîññèÿ E-mail address : from_dasha Ðàáîòà ïîääåðæàíà ÐÔÔÈ (ãðàíò 16-31-00138). 1