Geometrilly Relizing Nested Torus Links mie ry ugust 18, 2017 strt In this pper, we define nested torus links. We then go on to introdue ell deomposition of nested torus links. Using the ell deomposition nd its resulting irle pking we prove tht nested torus links n e geometrilly relized vi regulr idel othedr in hyperoli spe. 1 Introdution link is hyperoli if nd only if its omplement dmits omplete hyperoli struture. This mens the omplement must dmit metri of onstnt urvture 1. Hyperoli polyhedr provide nother wy to desrie hyperoli strutures on link omplements. Strting with the link omplement, one n find ell deomposition sliing the link omplement, so to define edges nd fes of n idel polyhedron in hyperoli spe. We then invoke ndreev s Theorem to result in irle pking ssoited with the idel polyhedr. ell deomposition is wy of ssigning 0, 1, 2, nd ells suh tht 0-ells hve no dimension, 1-ells re in one dimension, 2-ells re 2-dimensionl, nd ells re -dimensionl. utting long 2-ells determines the fes of the -ells, whih eome the idel polyhedr in H. ell deomposition, together with gluing instrutions, relizing link omplement s hyperoli must stisfy Poinré s Polyhedrl Theorem, see [1]. In [], hmpnerkr, et l use tetrhedrl deomposition to determine volume ounds for Dehn fillings of nested torus links -lled twisted torus links. Here we seek to etter understnd the geometry of nested torus links y proving they hve n othedrl geometri reliztion. The proof of Theorem in [4] sys if tht we n reognize n othedrl ell deomposition y looking t its resulting irle pking. 1.1 Torus Links efore defining nested torus links, we introdue torus links nd their nottion. Torus links re those whih lie on the surfe of n unknotted torus in -Spe, suh tht there re no self-intersetions of the link on the surfe of the torus. Suh links re lssified with the nottion (p, q), where p is the numer of times the link ompletes 1
D p-1 } 1 pth root D 1 p-1 1 σ 1 σ 2 σp-1 } q pth root Figure 1: The δ q p nested torus link full revolution round the longitude of the torus, nd q is the numer of times the link winds round the meridin. Note tht with simple isotopy of the torus, the (p, q) link is equivlent to the (q, p). The numer of omponents of the (p, q) link is the gd(p, q), see [2]. Thus when p nd q re oprime, the link is of one omponent - true torus knot. Following the nottion from [], torus links re relized s losed rids hving the form (σ 1 σ 2 σ p 1 ) q where δ p = σ 1 σ 2 σ p 1 is pth root of the full twist,, 2 p, of p strnds. Note tht δ p p = 2 p. 2 Nested Torus Links Nested torus links re the result of dding p 1 rossing irles, 1, 2,, p 1 over the (p, q) link suh tht 1 enompsses the entire losed rid, 2 enompsses the right most p 1 strnds, nd so on, until p 1 rosses over only the two right most strnds, see Figure 1. s onvention, we will lwys nest in this wy - so tht the inner most rossing irle enompsses only the right most strnds. The losed rid word for the (p, q) torus link is δ q p, so to differentite etween the torus link nd its nested prtner, the nottion for nested (p, q) torus link will herefter e δ q p. ssoited to eh rossing irle k is rossing disk,, perpendiulr to the plne of projetion. For simpliity, let ll e oplnr. It is importnt to note tht is thrie-puntured sphere. The inner most rossing disk,d p 1 is puntured twie y knot strnds nd one y the rossing irle p 1. In generl, is puntured y single knot strnd, k 1, nd k. Figure 1 demonstrtes generl δ q p. 2.1 Homeomorphisms nd quivlene of Nested Torus Links Our ultimte gol is to show nested torus links re othedrl. euse isotopi nd homeomorphi links hve the sme geometri struture, we use them to restrit the 2
() δ 1 5, The (5, 1) nested torus link () uliden refletion of δ1 5 ; the (5, 1) twisted torus link, lso ( δ 1 5 ) 1 = δ 5 1 Figure 2: quivlene of links vi uliden Refletion numer of ses we need to onsider. There re ertin llowle moves to desrie n equivlene vi homeomorphism etween nested torus links. Refletion Refletion is true uliden isometry. Therefore if there exists refletion etween two nested torus links over plne in the emedding spe (IR ), then the two nested torus links re homeomorphi. Figure 2 shows tht (p, q) (p, q), or in rid nottion: δ q p = (σ 1 σ 2 σ p 1 ) q (σ 1 σ 2 σ p 1 ) q = δ p q where δ p q is nested on the right nd δ q p is nested on the left. Isotopies Two links re isotopi if one n simply pull the strnds of the first link so tht it oinides with the seond, without utting, gluing, or hnging the sign of rossings. These re used to define topologil equivlene etween links. Full Twists In the disussion of fully ugmented links, it is known tht sliing long rossing disks nd regluing with full twist yields homeomorphi links. In the se of nested torus links, the sme is true over ny given rossing disk. Full twists re in the form δp, p or (p, p). Therefore, given nested full twist, δp p, we n pply the homeomorphism in whih the inverse of full twist is inserted: δ p pδ p p = (σ 1 σ 2 σ p 1 ) p (σ 1 σ 2 σ p 1 ) p = (σ 1 σ 2 σ p 1 ) 0 = δ 0 p In the ove nottion, we see tht inserting the inverse of full twist to full twist undoes eh of the rossings in the rid. Generlly, this mens δ 0 p δ p p. Furthermore, we n undo ny existing full twist in given δ q p nested torus link. Therefore, we know δ q p δ p qmodp. onsequently, let q < p for δq p.
Lemm 2.1. Up to homeomorphism, for δ q p, 0 < q p/2. Proof. We show this y proving if q + z = 0modp, then δ p q nd δ p z re homeomorphi. Rell tht q + z = 0modp if nd only if q + z = np for some n. Strting with δ p q = (σ 1 σ 2 σ p 1 ), pply the inverse of full twist n times. This is the omposition of n full twist homeomorphisms, so it is itself full twist homeomorphism.thus we hve: δ p q δ q np p ut q np = z, so we hve The homeomorphism from ell Deomposition δ q p δ q np p p δ z δ z p to δ p z is refletion. Throughout this setion, we prove tht nested torus links, δq p re othedrl for p nd q 1. Note tht δ 0 p ontins n emedded nnulus, regrdless of p, nd is thus not hyperoli. dditionlly if, p <, Lemm 2.1 sys q 1. If q = 0, then the link is lerly not hyperoli. ut when p < nd q = 1, the link lso ontins n emedded nnulus nd is thus not hyperoli. Furthermore, up to homeomorphism, proving tht nested torus links re otrhedrl for 1 q p/2 is suffiient for proving the sme result for ll q 1, y Lemm 2.1..1 Preliminries The results from [4] generlize the ell deomposition of hlf twist over singe rossing disk, see Figure 4, to one involving multiple strnds. In the se of nested torus links, we will gin generlize the ell deomposition for hlf twist over single rossing disk, ut only over suset of rid strnds. dditionlly, we will rell the geometry of p euse it will e illustrtive in understnding the geometry of δ q p. ells Rell tht in ell deomposition, 0-ells re end points for 1-ells, 1-ells ound 2-ells, nd 2-ells ound -ells. In the se of ell deomposition for fully ugmented link omplements, we ssign 1-ells s the intersetion of the rossing disk plne with the plne of projetion. The end points of these segments re knot nd rossing strnds, or puntures in the link omplement. Therefore, there re no 0-ells in the mnifold; these eome idel verties in hyperoli spe. The 2-ells re plnr regions, inluding twisted nds tht live long the plne of projetion, nd the -ells re spe like regions ove nd elow the plne of projetion. The plne of projetion splits spe so tht we hve 2 -ells, P + nd P. The 2-ells ounding P + re those visile from δ z p 4
k Figure : Flttened ell deomposition of n untwisted rossing disk ove. Similrly the 2-ells visile from elow re ssoited with P. ell Deomposition Over Single rossing Disk ell deomposition over untwisted rossing disks is well understood, through the disussion of fully ugmented links. s usul, there re no 0-ells, the 1-ells re the intersetion of the rossing disk with the plne of projetion, nd there re two -ells split y the plne of projetion. The 2-ells re determined so tht utting long them yields the oundry fes of the two -ells. Figure illustrtes tht the -ell P + is ounded y the plnr regions,,,, nd, nd twie y - one from the front nd one from the k. refletion of the sme piture ours for P. The ell deomposition over single hlf-twisted rossing disk, desried in [4], is similr to tht over n untwisted rossing disk. However, the hlf twist hnges the gluing instrutions for 2-ells, s illustrted in Figure 4, so tht P + is ounded y D k on the front nd on the k. Tht is, euse of the hlf twist, D k is visile from the -ell ove the plne of projetion. For this reson, it is helpful to look t the 2-ells for P + from the front nd k seprtely, keeping gluing instrutions onsistent. similr piture desries the ell deomposition on P. See [4] for more detils. ell Deomposition of Nested Hlf Twists Let us rell the generl struture of suh nested hlf twist, using the exmple 5. Figure 5 illustrtes the hlf twist over the entire rossing region, omposed of the su rossing disks,, F, G, nd H. Looking t the digrm from the front, onsider the inner most puntures of the rossing disk - single strnd nd the outer most punture of the rossing disk F. There is hlf twist over these two puntures. The sme is true etween the puntures of F, G, nd H. Tht is, there is hlf twist on the front side of eh rossing disk. Looking t the digrm from the k, eh of the rossing disks ly flt. Note tht in the gluing instrutions for the ell deomposition illustrte this niely, see Figure 5. We see,f,g, nd H on the front, orresponding to hlf twist. On the k, where the two ells re glued flt, we see, F, G, nd H. [4] Understnding Nested Torus Links 5
k Front k Figure 4: ell deomposition on hlf twisted rossing disk D k D k () 5 ell Deomposition () 5 Figure 5: Understnding the 5 link 6
In order to pply the ell deomposition of hlf twisted rossing disks to the more generl se of the nested torus link, we first isotope δ q p so to etter desrie gluing instrutions for 2-ells. Lemm.1. There is n isotopy of δ q p suh tht the rid n e viewed s hlf-twist over the first q rossing disks on the front, nd hlf twist over the lst q 1 rossing disks on the k. Proof. Strt with δ q p. Note tht the su-rid onsisting of the left-most q strnds is in the form (q, q) nd is thus full twist. The left q strnds ross over the lst p q strnds, so tht the left q strnds end up the right q strnds on the k. Isotope the lst hlf of the full twist on the left q strnds round to the k of the rossing disks. On this side, this represents hlf twist over the lst q strnds. euse the lst two strnds punture only the inner most rossing disk, this orresponds to hlf-twist over the lst q 1 rossing disks on the k. Figure 6 demonstrtes this isotopy for δ 2 5, nd the generl δ q p. q p q p Figure 6: Left: The δ 5 2 nested link fter the isotopy from Lemm.2. Middle: The front opy of δ p q fter Lemm.2. Right: The k opy of δ p q fter the sme isotopy..2 Nested Torus ell Deomposition Theorem.2. The nested torus link, δ q p is othedrl for p nd q 1 Proof. s forementioned, Lemm.1 results in Figure 6. euse we re deling with hlf twisted rossing disks, it is helpful to use this piture to determine the ell deomposition for δ q p, keeping onsistent gluing instrutions. dditionlly, we look t the ell deomposition of P +. The proof for P is nlogous. 1-ells euse eh rossing disk is thrie puntured sphere, the resulting ell deomposition will hve n even numer of tringulr rossing disk fes, eh ounded y 1-ells. The 1-ells ounding rossing disk re denoted 1 k, 2 k, nd k. For ll 7
D 1 1 2 1 1 1 2 2 i i D p-1 2 1 1 1 1 p p p 1 k 1 D' p-1 D' 1 Figure 7: 1-ells of δ q p. vlues of k, k is to the right of the losed rid euse of nesting. See Figure 7. 2-ells First preform the isotopy from Lemm.1. The 2-ells re determined y grouping like strnds on hlf-twisted nds, nd onsidering the front nd k seprtely. See lels W, X, Y nd Z in Figure 8. Regions ontining mny knot strnds deompose into mny seprte 2-ells.The hlf-twisted nd, denoted region Y in the Figure 8, ontins q strnds. s result, there re q 1 distint 2-ells grouped together in this region. These re ounted y the numer of plnr spes etween strnds. Similrly, the flt nd, denoted y region W hs p q strnds, nd p q 1 distint 2-ells within this region. The regions denoted X, V nd Z eh ontin extly one 2-ell euse they re not slied y knot strnds. Now euse ll twisting ours ner the rossing disks, wy from the rossing disks, ll 2-ells lie plnr - or untwisted. s we pproh rossing disks on the front, the left q strnds lie on hlf-twisted nd; the right p q strnds lie on flt nd. The 2-ells etween these nds re s in the se of single hlf twist pplied to this more generl sitution. Note tht the first q rossing disks hve hlf twist over their rossing disks on the front. Therefore, y the ell deomposition desried in [4], D 1 D q re visile from P + on the front. On the k, the rossing disks D q+2 D p 1 re visile from P + for the sme reson. ll other rossing disks re flt. Figure 8 demonstrtes regions of two ells, inluding their gluing instrutions over rossing irles. Front k v w x y z z y x w v Figure 8: 2-ell regions of δ q p. Flttening the ell Deomposition Shrinking the rossing strnds to idel verties results in Figure 9. Note tht Region Y ontins q 1 2-ell fes, nd Region W ontins p q 1 2-ell fes, s previously explined. h of these regions must 8
ontin t lest one 2-ell ounded y extly three 1-ells euse eh region hs two strnds punturing the inner most rossing disk. These 2-ells re tngent to extly three other 2-ells, not inluding rossing disks. y v i x w D 1 ' D 1 2 p 1 ' Dp 1 j D q + 1 i ' Dq+2 1 q+2 + z 1 q + 1 ' D q D q + 1 j Dp 1 2 p 1 Figure 9: flttended 2-ells of δ q p. irle Pking Finlly, we shrink the knot strnds to result in irle pking, Figure 10. h 2-ell, not inluding those from rossing disk shrinks to irle. The rossing disks shrink to n even numer of tringulr fes. The 2-ell region Z is tngent to eh 2-ell in regions X, W, nd V, nd only the outermost 2-ell of region Y. This is the outside fe of irle pking. The 2-ell in region X is tngent to every other 2-ell, so eh irle in the resulting irle pking must e tngent to the irle leled X. h region W nd Y ontins 2-ell tht deomposes to irle tngent to extly three other irles, euse oth regions ontin tringulr 2-ell. ording to the proof of Theorem in [4], we n reognize irle pking s othedrl if the it is the result of dding mutully tngent irle etween three existing mutully tngent irles; the orresponding polyhedron is the result of ppending n othedron onto n existing polyhedron. Through this, we see tht the resulting irle pking ssoited to ny δ q p orresponds to n idel polyhedron formed y ppending othedr to n existing polyhedron. Therefore, the link omplement for δ q p is othedrl. Figure 11 illustrtes the othedrl ell deomposition of δ 2 5. 9
Z W } p-q-1 X W Y Y (q-1 fes) v Figure 10: irle pking ssoited to δ q p. D4' D D 1 D 2 D D 4 D 4 D 2 D D 1 D' 4 D' D' 2 D' 1 D1 D2 D4' D1' D'2 D1 D D1' D2' D4 D2 D D D4 (d) irle pking () δ 2 5 efore isotopy () δ 2 5 fter isotopy () ell deomposition Figure 11: ell deomposition for δ 2 5 4 Open Questions How does knowing tht nested torus links re othedrl improve volume ounds for their Dehn fillings, twisted torus links? 5 knowledgments I would like to extend speil thnks to Dr. Rollnd Trpp for his support nd enourgement throughout my explortion of this prolem. I would lso like to thnk Dr. orey Dunn nd Dr. Trpp for their dedition to undergrdute reserh nd their outstnding efforts in sponsoring the liforni Stte University Sn ernrdino RU progrm. This projet ws funded oth y NSF grnt DMS-1461286 nd liforni Stte University Sn ernrdino. 10
Referenes [1] F. onhon. Low-Dimensionl Geometry: from uliden surfes to hyperoli knots, volume 49. merin Mthemtil Soiety, 2009. [2] M.. ush, K. R. Frenh, nd J. R. H. Smith. Totl linking numers of torus links nd klein links. Rose-Hulmn Undergrdute Mthemtis Journl, 15:72 92, 2014. Pulished online. []. hmnerkr, D. Futer, I Kofmn, W. Neumn, nd J. S. Purell. Volume ounds for generlized twisted torus links. Mthemtil reserh letters, 18:1097 1120, 2011. [4] J. Hrnois, H. Olson, nd R Trpp. Hyperoli tngle surgeries nd nested links. preprint. 11