Slope Lengths for 2-Bridge Parent Manifolds. Martin D. Bobb
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1 Cliorni Stte University, Sn Bernrino Reserh Experiene or Unergrutes Knot Theory Otoer 28, 2013
2 Hyperoli Knot Complements Hyperoli Knots Deinition A knot or link K is hyperoli i hyperoli metri n e ple on its omplement, S 3 \ K. Figure : The igure-8 knot is the hyperoli knot whose omplement hs the smllest known totl volume.
3 Hyperoli Knot Complements Universl Covers A hyperoli 3-mniol, M, suh s hyperoli knot omplement, hs universl over in H 3, hyperoli spe. This mens tht H 3 my e tessellte y iel hyperoli polyher tht glue together to rete the mniol M. Figure : The upper-hl spe hyperoli metri n n iel hyperoli otoheron.
4 Hyperoli Knot Complements Covering the Torus Both gluings elow will proue the milir torus ( oughnut sure). B A A B Figure : The over on the right is universl over n inues metri on the torus sure.
5 Dehn Fillings A Dehn Filling is the t o gluing soli torus into toroil omponent missing rom knot omplement. We my irst use homeomorphism on the sure o the soli torus to inue ull twists on strns pssing through the torus hole. Figure : Perorming Dehn illing on the re omponent twists the two lk strns n removes the re omponent.
6 Dehn Fillings Volume Chnge Perorming Dehn illings on hyperoli mniol reues the volume o the mniol. This reution in the volume is relte to the slope length ssoite to the Dehn illing. This is the length to whih one meriin o the soli torus will e glue on the ounry omponent o the hyperoli mniol. Meriins on the soli torus will lwys e glue to n integer numer o meriins n longitues on the ounry omponent. These slope lengths re oun in the universl over o the mniol.
7 Dehn Fillings Lower Boun on Volume Chnge Futer, Klginni, n Purell presente lower oun on the hnge in volume o mniol, V (M), ue to Dehn illings. This oun is ( ( ) ) 2π 2 3/2 Vol(S 3 \ K) 1 V (M). l min Where l min is the minimum slope length use over ll Dehn illings.
8 A 2-Brige Knot A 2-rige link hs n emeing with only two lol mxim (n minim).
9 Prent Mniols We sy tht mniol M is prent mniol or nother mniol M i numer o Dehn illings on M will proue M 1 2 Figure : The Borromen rings omplement ts s prent mniol or ny 2-rige knot with two twist regions.
10 Borromen Rings Universl Cover To in universl over or the Borromen rings omplement, we perorm ell eomposition on the spe. R G e B
11 Borromen Rings Universl Cover R G B e e e e R G B e e e e Figure : Cell eomposition on top (let) n ottom (right) portion o the Borromen rings omplement.
12 Borromen Rings Universl Cover Gluing these two otoher together gives unmentl omin or the Borromen rings omplement. e e e e
13 Fining Slope Lengths Slope lengths lie on toroil neighorhoos o link omponents. The pre-imges o these neighorhoos in the universl over re olletion o horolls, Eulien spheres tngent to the Riemnnin sphere t ininity in H 3. A horoll is si to e entere t its point o tngeny.
14 Fining Slope Lengths To in the lengths o slopes on horoll neighorhoos, we ompre the tesseltion present on its sure to the eets o our ell eomposition on the omponent neighorhoo. Figure : A toroil omponent neighorhoo n its pre-imge, tessellte y squres.
15 Funmentl Domin or the Torus on Horoll Ater eveloping the tesselltion on the horoll, we in this unmentl omin or the torus: The meriin n longitue re hightlighte purple n green respetively. Using the metri inue on the horoll, we in tht they hve length 2 n 2 2 respetively.
16 Improving Bouns Thus, or 2-rige knot, K, with two tngle regions, eh region hving t lest m 2 ull twists, the volume ( ( ) Vol(S 3 2π 2 3/2 \ K) 1 2v 8. 2m 2 + 8) where v 8 = is the volume o n iel otoheron. This is n pplition o the oun presente y Futer, Klginni, n Purell.
17 Generlize 2-Brige Prent Mniols A prent mniol or ny 2-rige knot omplement n e uilt rom Borromen rings omplements y the gluings isusse erlier: glue
18 Generlize 2-Brige Prent Mniols Universl Covers I we ut n reglue the Borromen rings omplement unmentl omin, we my expose the thrie punture spheres long whih we glue: e e e e
19 Generlize 2-Brige Prent Mniols Universl Covers Reglue, with the piee on the right glue eneth tht on the let. Figure : Ege gluings re omitte here or simpliity.
20 Generlize 2-Brige Prent Mniols We my now glue long these expose sures to in unmentl omins or more generl prent mniols. This lso inues gluings on the portion o the unmentl omin glue eneth. 1 2 n n-1 n 1 2 n-1 n n n n Figure : Ege gluings re omitte here or simpliity.
21 Open Questions 1 Given the unmentl omin rom the lst slie, n we in slope lengths or more generl prent mniols? 2 Wht ptterns will emerge upon exploring these slope lengths? 3 Cn we pply the sme methoology to other lsses o knots?
22 Aknowlegements I woul like to knowlege Dr. Rollie Trpp or his support, enourgement, n help uring this reserh n Dr. Corey Dunn or mking this reserh experiene possile. This reserh ws jointly une y 2013 NSF-REU grnt DMS , n y Cliorni Stte University, Sn Bernrino.
23 Reerenes 1 Colin C. Ams, Thrie-Punture Spheres In Hyperoli 3-Mniols. Trnstions O The Amerin Mthemtil Soiety, Volume 287, Numer 2, Ferury Dvi Futer, Estrti Klginni, Jessi S. Purell,Dehn Filling, Volume, An The Jones Polynomil.Journl O Dierentil Geometry, Dvi Futer, Jessi S. Purell, Links with no exeptionl surgeries. Commentrii Mthemtii Helvetii, 82, Jessi S. Purell, Slope lengths n generlize ugmente links. Communitions In Anlysis An Geometry, Volume 16, Numer 4, Hyperoli Geometry O Multiply Twiste Knots. rxiv: v2, June 2009.
24 Thnk You!
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