Recent progress of various Volume Conjectures for links as well as 3-manifolds (Intelligence of Low-dimensional Topology)
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(2) 57. (2, 2m) ‐torus links [26], Borromean rings [21], twisted Whitehead links [60] and Whitehead [58]. Actually progress towards hyperbolic knots except the figure‐eight knot 4_{1} and knot 5_{2} were relatively less. Recently T. Ohtsuki [48] obtained the whole asymptotic expansion of Kashaevs invariants (colored Jones polynomial J_{n} evaluated at e^{\frac{ $\pi$\sqrt{-1} {N} ) of knot 5_{2} By a similar fashion procedure, Ohtsuki‐Yokota [52], T. Ohtsuki [49] and T. Takata [57] obtained asymptotic expansion of Kashaevs invariants of hyperbolic knots with 6, 7 and 8 crossings ( 8_{6} and 8_{12} ) respectively. chains. .. Some of my recent work. on. Volume. Conjectures. divided to two research directions. In this survey, in these two directions.. we. and related. will discuss. topics. some. be roughly developments. can. recent. appreciates comments from Renaud Detcherry, Giovanni Felder, Efstra‐ Kalfagianni, Kefeng Liu, Jun Murakami, Tomotada Ohtsuki, Tian Yang and Sheng‐ mao Zhu. The author is supported by the National Center of Competence in Research The author. tia. SwissMAP of the Swiss National Science Foundation.. 2. Volume. Conjectures. for Reshetikhin‐Turaev and Turaev‐Viro invariants. of 3‐manifolds Volume. 2.1. Conjectures. for Reshetikhin‐Turaev and Turaev‐Viro invariants. of 3‐manifolds evaluated at roots of In. [11], Chen‐Yang. unity. q(2)=\displaystyle \exp(\frac{2 $\pi$\sqrt{-1} {r}). first extended the Turaev‐Viro invariant from closed 3‐manifolds to. 3‐manifolds with cusps (orientable case by Benedetti‐Petronio [2]) or even with totally geodesic boundary. Recall that Turaev‐Viros original construction gives rise to real valued invariants for closed 3‐manifolds and 2 + 1. TQFTs for 3‐cobordisms containing a link constructions, they triangulations, meaning the vertices of the triangulations are inside the manifolds and the cobordisms, and in the case of cobordisms, the edges on the boundary are from the edges of the triangulations. The difference in our construction [11] is that, instead of using the usual triangulations, we use ideal triangulation of a 3‐manifold with non‐empty boundary. inside. In all of their. e\displaystyle\frac{$\epsilon\pi$\sqrt{-1} {r}. the usual. integer s where (r, s)=1 Recall that complex‐valued invariants of oriented \{$\tau$_{r}(M;q)\} closed 3‐manifolds with q This provided a mathematical construction of the q(odd) 3‐manifold invariants (evaluated at q(1) only) introduced by Witten using Chern‐Simons action. Using skein theory, Lickorish [38, 39] redefined the Reshetikhin‐Turaev invariants with q q(odd) and Blanchet‐Habegger‐Masbaum‐Vogel [3] extended them to q(2) Let. q(s). use. denotes the roots of. unity. for. the Reshetikhin‐Turaev invariants =. =. e^{\frac{2 $\pi$\sqrt{-1} {r}. an. .. ,. are. .. =. ,. for odd. r. .. In. [11], Chen‐Yang proposed. the. following. Volume. Conjecture.. (Volume Conjectures for Reshetikhin‐Turaev and Turaev‐Viro invari‐ [11]). Let M_{1} be a closed oriented hyperbolic 3‐manifold and let $\tau$_{r}(M_{1};q) be its Reshetikhin‐Turaev invariants. Let M_{2} be a compact 3‐manifold (cusped or with to‐ tally geodesic boundary). Then for r running over all odd integers and q=q(2)=e^{\frac{2 $\pi$\sqrt{-1}}{r}}. Conjecture. 2.1. ants, Chen‐Yang.
(3) 58. [n]=\displaystyle \frac{q^{n}-q^{-n} {q-q-1}. under the convention. ,. \displaystyle \lim_{r\rightar ow+\infty}\frac{4 $\pi$}{r-2}\log($\tau$_{r}(M_{1};q) CS(M_{1}). where. and with. a. suitable choice. is the usual Chern‐Simons invariant. of M_{1} multiplied by 2$\pi$^{2} =. (also Turaev‐Viro). Turaev. we. have. ,. and. Vol(M2).. fact, Wittens Asymptotic Expansion Conjecture (WAE). turns out to be of. the arguments,. (mod \sqrt{-1}$\pi$^{2} ),. \equiv Vol (M_{1})+\sqrt{-1}CS(M_{1}). \displaystyle \lim_{r\rightar ow+\infty}\frac{2 $\pi$}{r-2}\log(TV_{r}(M_{2};e^{\frac{2 $\pi$\sqrt{-1} {r} ) In. of. invariants evaluated at usual root of. considers Reshetikhin‐. unity. q(1)=e\displaystyle \frac{ $\pi$\sqrt{-1} {r}. This. For past 25 years, people have thought Reshetikhin‐Turaev and Turaev‐Viro invariants evaluated at other roots of unity should. only polynomial growth. w.r. \mathrm{t}.. r. .. have similar. asymptotic behavior which is also polynomial growth w.r. \mathrm{t}. r. not only corrects a long existing wrong feeling about Reshetikhin‐Turaev and Turaev‐Viro invariants but also largely extends the original Vol‐ ume Conjecture (Kashaev‐Murakami‐Murakami) from knot complements in S^{3} to all kinds of hyperbolic 3‐manifolds, no matter they are closed, cusped or even with totally geodesic Thus. Chen‐Yangs conjecture. boundaries.. Some further. by. developments of Chen‐Yangs Volume Conjectures. have been announced. various mathematicians in several recent international conferences. results \bullet. are. listed. as. Ohtsuki[50] first generalize our Volume Conjecture for Reshetikhin‐Turaev in‐ variants at q(2) =e\displaystyle \frac{2 $\pi$\sqrt{-1} {r} to a full asymptotic expansion conjecture (physics flavour conjecture by D. Gang‐M. Romo‐M. Yamazaki[20]) and then he proved our Volume Conjecture. for. a. series cases, the closed hyperbolic 3‐manifolds S^{3} along the figure‐eight knot 4_{1}.. M_{K} obtained by. surgery in. Takata[51] recognized the Reidemeister asymptotic expansion.. T. Ohtsuki‐T.. above full \bullet. arXiv. These. T.. integer Dehn \bullet. or on. follows. torsion term. appearing. in the. Detcherry‐E. Kalfagianni‐T. Yang[14] proved our Turaev‐Viro Volume Conjecture cases of hyperbolic cusped 3‐manifolds S^{3}\backslash 4_{1}, S^{3}\backslash Borromean rings via establish‐ ing a relation involving Turaev‐Viro invariants of link complements in S^{3} and certain sum of colored Jones polynomials of that link. R.. for. \bullet. R.. Detcherry‐E. Kalfagianni[13] first. of the Turaev‐Viro invariants at the Gromov. norm. of. established. q(2)=e\displaystyle \frac{2 $\pi$\sqrt{-1} {r}. a. that. relation between the. Chen‐Yang 3‐manifolds. Furthermore, they obtained. asymptotics. considered in a. [11]. and. lower bound for the. Gromov norm of any compact, oriented 3‐manifold with empty or toroidal boundary, in terms of the Turaev‐Viro invariants. They also proved Turaev‐Viro Volume Con‐. jecture for Gromov. norm zero. several torus knots / links such. Yangs original. paper. links as. [11]). Finally they constructed infinitely families. for Turaev‐Viro invariants with ume. conjecture. (knots proved by Detcherry‐Kalfagianni‐Yang[14], knot, Hopf link, link T(2,4) proved in Chen‐. Trefoil. of 3‐manifolds. exponential growth predicted by Chen‐Yangs Vol‐.
(4) 59. Usual. Figure \bullet. (compact) Edges of. 1:. tetrahedron. a. Kolpakov‐J. Murakami[34] formulated. A.. Kirillov‐Reshetikhin invariants at \bullet. tetrahedron. the. corresponding Volume Conjecture for. q(2)=e\displaystyle \frac{2 $\pi$\sqrt{-1} {r}. Chen‐Yang. that. considered in. [11]. Q. Chen‐J. Murakami[10] proposed a Conjecture for dominant term (Volume) and secondary term (Gram matrix of the tetrahedra) in asymptotics of the quantum 6j symbol. (at. cases. at. least. q(2) =e^{\frac{2 $\pi$\sqrt{-1} {r} one. that. Chen‐Yang. Q. Chen‐J. Murakami [10]. also. asymptotics of quantum 6j. symbol. proposed at. a. [11]. considered in. of the vertex of the tetrahedra is ideal. Conjecture. q(2)=e^{\frac{2 $\pi$\sqrt{-1} {r}. and. proved majority Furthermore, symmetric property of. Ultra‐ideal.. or. for. a. observed from. big. cancellations.. (Symmetry of asymptotics of quantum 6j symbols, Chen‐Murakami hyperbolic tetrahedron and $\theta$_{a}, $\theta$_{b}, $\theta$_{c}, $\theta$_{d}, $\theta$_{e}, $\theta$_{f} be dihedral angles at edges a, \cdots, f in above Figure. Let a_{k}, b_{k}, \cdots, f_{k} be sequences of non‐negative half integers satisfying. Conjecture. [10]).. 2.2. Let T be. a. \displaystyle \lim_{k\rightar ow\infty}\frac{4 $\pi$}{k}a_{k}= $\pi-\theta$_{a}, \lim_{k\rightar ow\infty}\frac{4 $\pi$}{k}b_{k}= $\pi-\theta$_{b}, \cdots , \lim_{k\rightar ow\infty}\frac{4 $\pi$}{k}f_{k}= $\pi-\theta$_{f} that the. so. odd r\geq 3. .. triplets (a_{r}, b_{r}, e_{r}). a_{r}'. Let. be. a. (a_{r}, d_{r}, f_{r}) (b_{r}, d_{r}, f_{r}) (c_{r}, d_{r}, e_{r}). ,. ,. sequence. ,. are. all. r ‐admissible. for. of non‐negative half integers satisfying. \displaystyle \lim_{k\rightar ow\infty}\frac{4 $\pi$}{k}a_{k}'= $\pi$+$\theta$_{a} so. of. that the. triplets (a_{r}', b_{r}, e_{r}) (a_{r}', d_{r}, f_{r}) ,. are r ‐admissible.. \left\{ begin{ar y}{l a_{r}&b_{r}&e_{r}\ d_{r}&c_{r}&f_{r} \end{ar y}\right\} \left\{ begin{ar y}{l a_{r}'&b_{r}&e_{r}\ d_{r}&c_{r}&f_{r} \end{ar y}\right\} and. with respect to. Then the r. are. asymptotic expansions. equal.. Remark 2.1. Since the volume function and the gram matrix of a tetrahedron are not of a dihedral angle, the above conjecture is true up to the. changed by switching the sign second leading term (by main. the third term for 2.2. Volume. some cases. Conjectures. theorem in. by. [10]).. numerical. for Reshetikhin‐Turaev and Turaev‐Viro invariants. of 3‐manifolds evaluated at roots of Now. we. ants at. have. q(2). .. new. Volume. We also checked that the coincidence of. computation.. Conjectures. unity q(odd). for Reshetikhin‐Turaev and Turaev‐Viro invari‐. So it is natural to ask whether there also exists Volume. Conjectures. for.
(5) 60. q(odd). Reshetikhin‐Turaev and Turaev‐Viro invariants at. .. example of Turaev‐Viro invariant of non‐orientable 3‐manifold N_{2_{1}} (Callahan‐ Hildebrand‐Weeks census) vanishes at roots of unity q(s) where r and s are both odd numbers and (r, s)=1 (required by a condition from the definition of the Turaev‐Viro in‐ variant). Numerical evidence shows that it is nonzero at q(s) and also goes exponentially large as r\rightarrow\infty when s is an odd number other than 1 but r is an even number. Take into account the above example it is natural to propose the following Volume Conjecture, In. fact,. the. ,. ,. 3‐manifold M with boundary (orientable or non‐orientable, totally geodesic boundary), for a fixed odd number s other than 1 and such that condition (*) TV_{r}(M, q(s))\neq 0 is satisfied, then we have. Conjecture or even. integer. 2.3. For any. with r. \displaystyle \lim \underline{s $\pi$}_{\log}|TV_{r}(M, q(s))|=Vol_{cpx}(M). r\rightaatiisfy rrow\infty(s)=1rs *) Remark 2.2. If. boundary,. we. TV_{r}(M, q(s)) \neq. could. change. .. r. 0 for all any. condition. r. integer. even. satisfy (*). . to. \mathrm{r}. and any 3‐manifold M with. is even. r. Remark 2.3. When M has cusps, as in [10], we consider ideal tetrahedral decomposition of M , and consider Turaev‐Viro invariant of this tetrahedral decomposition. When M has. totally geodesic boundary, we consider the singular 3‐manifold obtained from M by col‐ lapsing each boundary component, and consider Turaev‐Viro invariant of a triangulation of this singular 3‐manifold. Similar. phenomenon. also. happens to the Reshetikhin‐Turaev invariants.. Turaev invariants of closed 3‐manifold M obtained from. K, RT_{r}(M, q(s)). and. (r, s). =. ,. vanishes at roots of. 1 and. (See Kirby‐Melvin. unity q(s) where ,. and. see. large. as. r\rightarrow\infty ,. r. The Reshetikhin‐. 4k+2 ‐surgery along and. s. are. a. knot. both odd numbers. Chen‐Liu‐Peng‐Zhu [7]). Numerical examples we tested are nonzero at q(s). also. evidence shows that Reshetikhin‐Turaev of certain and also goes exponentially is an even number.. a. when. s. is. an. odd number other than 1 but. r. Take into account the above. example. it is natural to propose the. following. Volume. Conjecture, Conjecture 2.4. For any closed 3‐manifold M (orientable or non‐orientable, or even with totally geodesic boundary), for a fixed odd number s other than 1 and such integer r that condition (**) RT_{r}(M, q(s))\neq 0 is satisfied, then we have \displaystyle \lim. r\rightaatisfy rrow\infty'((r,**) s)=1rs Remark 2.4. If manifold M ,. we. \underline{2s $\pi$}_{\log}(RT_{r}(M, q(s) ) =Vol_{cpx}(M). RT_{r}(M, q(s)) \neq could. (mod \sqrt{-1}$\pi$^{2}\mathbb{Z} ).. r. change. 0 for all any. condition. r. even. integer. satisfy (**). . to. \mathrm{r}. r. and any 3‐manifold closed is even.
(6) 61. Here is. a. summary of various results.. Volume. Conjectures. for 3‐manifolds with. or. without boundaries and. quantum 6\mathrm{j} symbols. 2.3. New Volume. Conjectures. in mathematics may indicate. new. physics. By many numerical checks and proofs of several non‐trivial examples of our Volume Conjecture, it is clear that Reshetikhin‐Turaev/Turaev‐Viro invariants evaluated at root of unity q(2) display a much deeper hidden relation to the hyperbolic geometry than those evaluated at usual roots of unity q(1) Connections of the original Volume Conjecture with physics were explored by Gukov [23], Dijkgraaf‐Fuji‐Manabe [15] and Witten [59] and the physics meaning of Reshetikhin‐ Turaev invariants evaluated at root of unity q(1) was the original quantum Chern‐Simons theory studied by Witten. So it is very natural to ask the physical meaning of the Reshetikhin‐Turaev / Turaev‐Viro invariants evaluated at the root of unity q(2) Due to the nega‐ tive quantum integer involved, their nature should be a non‐unitary physics theory, which looks a bit crazy, but it is confirmed by many top mathematical physicists such as G. Felder, R. Kashaev, N. Reshetikhin and E. Witten etc. Drastic cancellation appears in the computation of the Turaev‐Viro invariants of hyperbolic 3‐manifolds evaluated at roots of unity q(2) when T. Yang and the author .. .. ,.
(7) 62. numerically check those examples in [11]. This means that one still obtain exponentially large identities of Turaev‐Viro invariants evaluated at roots of unity q(2) even after those drastic cancellations. This never happens to the original Volume Conjecture, so we believe our Volume Conjecture display a very unexpected/wild nature of quantum invariants, which make its corresponding physics meaning more mysterious. The above new discovery shed some new lights on postulating a new quantum Chern‐ Simons theory corresponding to the Reshetikhin‐Turaev invariants evaluated at non‐ conventional roots of unity such as q(2) etc. we expect that such a potential physics explanation will have a very wide application just like Wittens quantum Chern‐Simons theory which corresponds to the Reshetikhin‐Turaev invariant evaluated at q(1) and will shed some new lights on the current study of High Energy Physics. 3. A. on. proposing. backgrounds. of. physics background of. the. Some. 3.1. The. vision. new. Volume. large. \mathrm{N}. Conjectures. duality. and LMOV. Conjectures. problem the author concerned can be traced back to the large N expansion of U(N) gauge field theories in 1974. It was discovered by Gopakumar and Vafa that the topological string theory on the resolved conifold is dual to the U(N) Chern‐Simons theory on S^{3} This striking duality means the partition functions of two different theories exactly agree up to all orders. The former one corresponds to the (open) Gromov‐Witten theory which is still in its infancy, while the latter one corresponds to quantum invariants of links and 3‐manifolds. The open large \mathrm{N} duality was established by Ooguri‐Vafa [53]. Quantum group invariants W_{V^{1},\cdots,V^{L} ^{\mathrm{g} (\mathcal{L})(q) of link \mathcal{L} were determined by representa‐ tions V^{ $\alpha$} of U_{q}(\mathrm{g}) the quantized universal enveloping algebra of \mathrm{g} A partition A can be labeled by the Young tableau which corresponds to an irreducible representation V_{A} for a specific \mathfrak{g} sl_{N} and there exists a two‐variable colored HOMFLY‐PT invariant s.t. Let \mathcal{P}^{L} denote all the W_{A^{1},\cdots,A^{L} ^{\mathrm{g} (\mathcal{L})(q, t) W_{A^{1},\cdots,A^{L} ^{\mathfrak{g} (\mathcal{L})(q, t)|_{t=q^{N} seminal work of \mathrm{t} Hooft. on. .. .. ,. =. ,. sets labeled. the. partition by Young partition function of \mathcal{L} is defined by. =W_{V_{A^{1} ,\cdots,V_{A^{L} }^{\mathrm{g} (\mathcal{L})(q). .. tableau. For each link \mathcal{L} , the type -A Chern‐Simons. Z^{SL}c(\displaystyle\mathcal{L};q,t)=\sum_{\not\supset_{\in\mathcal{P}^{L} W^{SL}(\mathcal{L};q,t)_{S}\not\supset(x)=\sum_{\mathrm{i}?\neq0}F_{\mathrm{i}? ^{SL}p_{\mathrm{i}^{7} where. s\not\supset(x). \displayst le\sum_{i=1}^{+\infty}(x_{i})^{n}.. By using. are. the. the free energy. the Schur. polynomials of partition \vec{A}. plethystic exponential as. follows. method. (due. to. =. (A^{1}, A^{L}). and p_{n}. Getzler‐Kapranov),. \displaystyle \log Z_{CS}^{SL}(\mathcal{K};q, t;x) = \sum_{A\in \mathcal{P} \sum_{d=1}^{\infty} \frac{f_{A}(\mathcal{K};q^{d},t^{d}) {d}s_{A}(x^{d}). .. we can. =. write.
(8) 63. on the large N duality, Labastida‐Marino‐Ooguri‐Vafa [36, 37, 53] conjectured amazing algebraic structure for the generating series (Chern‐Simons partition function) of colored HOMFLY‐PT link invariants and the integrality of the infinite family of new topological invariants.. Based. an. Conjecture exists. a. 3.1. (LMOV Conjecture, Labastida‐Marino‐Ooguri‐Vafa, 2000‐2002).. knot invariant. P_{B}(\displaystyle \mathcal{K};q, t)\in\frac{1}{(q-q-1)^{2} \mathbb{Z}[(q-q^{-1})^{2}, t^{\pm 1}]. f_{A}(\displaystyle \mathcal{K};q, t)=\sum_{|B|=|A|}P_{B}(\mathcal{K};q, t)M_{AB}(q) M_{AB}(q). where. =. \displayst le\sum_{|$\mu$|=A|}\frac{$\chi$_{A}(C_{$\mu$}) \chi$_{B}(C_{$\mu$}){3$\mu$}\prod_{j=1}^{\el($\mu$)}(q^{$\mu$_{j} -q^{-$\mu$_{j}). ,. and $\chi$_{A},. There. s.t.. ,. and ồ $\mu$. C_{ $\mu$}. are. character,. conjugacy class and multiplicity labelled by Young tableau respectively.. original LMOV conjecture describes a very subtle structure of Z_{CS}^{SL}(\mathcal{L};q, t) which proved by Liu‐Peng [41]. Mathematically, the LMOV conjecture confirms that colored HOMFLY‐PT invariants also have integrality, symmetry of q pole order structure of q-q^{-1} just like classical HOMFLY‐PT polynomials. Physically, these integer coefficients correspond to the BPS states on Calabi‐Yau 3‐folds. The. ,. was. ,. Orthogonal LMOV conjecture. 3.2. L. Chen and the author. [6]. gave. a. mathematically rigorous formulation of orthogonal. LMOV conjecture dealing with the colored Kauffman invariants by using the theory of the Birman‐Murakami‐Wenzl algebras. By using the cabling technique, we obtained [6] \mathrm{a} uniform formula of the colored Kauffman. polynomial for all torus links. Then we were able of cases this interesting orthogonal LMOV conjecture. The topological [6] string side of the large N duality of the orthogonal LMOV conjecture is the open Gromov‐ Witten theory of orientifolds developed purely by physicists, while its math is still in its infancy. to prove. many. Congruence. 3.3. skein relations for colored HOMFLY‐PT invariants. The reformulated colored HOMFLY‐PT invariant. \check{\mathcal{Z} _{\vec{ $\mu$} (\mathcal{L};q, t). is defined. \displayst le\check{\mathcal{Z}_{\vec{$\mu$}(\mathcal{L};q,t)=[\vec{$\mu$}]\sum_{A^ $\alpha$}\prod_{$\alpha$=1}^{L}$\chi$_{A^ $\alpha$}($\mu$^{$\alpha$})\overline{W}_{\vec{A}(\mathcal{L};q,t) where. \overline{W}_{\vec{A} (\mathcal{L};q, t). is the. framing dependent. and $\chi$_{A^{ $\alpha$}} is the character of the irreducible A^{ $\alpha$}. In use. particular,. the notation. \mathcal{Z} in the above. suppose. \vec{$\mu$}. \check{\mathcal{Z} _{p}(\mathcal{L};q, t). =. ((p), (p)). as. ,. colored HOMFLY‐PT invariant colored by \vec{A} representation indexed by the Young tableau with L. row. partitions (p) for ,. p \in. \mathbb{Z}_{+}. .. We. to denote the reformulated colored HOMFLY‐PT invariant. simplicity. Although the definition of \check{\mathcal{Z} _{\vec{ $\mu$} (\mathcal{L};q, t) seems complicated definition, it has a simpler form than the colored HOMFLY‐PT invariant. q , t ) for.
(9) 64. W_{\vec{A}}(\mathcal{L};q, t). in the the HOMFLY‐PT skein. and it is natural to instead of. \overline{W}_{\vec{A} (\mathcal{L};q, t). study. \check{\mathcal{Z} _{\vec{ $\mu$} (\mathcal{L};q, t). has nice. properties. the reformulated colored HOMFLY‐PT invariant. \check{\mathcal{Z} _{\vec{ $\mu$} (\mathcal{L};q, t). theory.. In. fact,. .. All classical knot invariants. can. be defined from. a. simple computational rule, the skein polynomials. relation. Under the above setups, classical skein relation for HOMFLY‐PT can be restated as follows. For any link \mathcal{L} , we have. \check{\mathcal{Z} _{1}(\mathcal{L}_{+};q, t)-\check{\mathcal{Z} _{1}(\mathcal{L}_{-};q, t)=\check{\mathcal{Z} _{1}(\mathcal{L}_{0};q, t) type I \check{\mathcal{Z} _{1}(\mathcal{L}_{+};q, t)-\check{\mathcal{Z} _{1}(\mathcal{L}_{-};q, t)=\{1\}^{2}\check{\mathcal{Z} _{1}(\mathcal{L}_{0};q, t) type where type I means self‐crossing in different link components. The. link component and type II. a. question whether quantum invariants share the perplexed people for quite a long time due. relation has. of quantum invariants. compute, and even the. of the. figure‐eight. means. crossing. in. property of certain skein. same. complexity of definition notoriously hard to arbitrary shape of Young. to the. The colored HOMLFY‐PT invariants case. II,. knot 4_{1} with. are. tableau is not established.. Inspired by studying the framed LMOV conjecture (a generalization of the original Conjecture mentioned in the last subsection), Chen‐Liu‐Peng‐Zhu [7] discovered a very interesting phenomenon called congruence skein relations, which means that skein relations hold for (reformulated) colored HOMFLY‐PT at certain roots of unity. LMOV. Conjecture 3.2 (Congruence skein relations for the colored HOMFLY‐PT invariants, Chen‐Liu‐Peng‐Zhu [7]). For any link \mathcal{L} and prime number p we have ,. \check{\mathcal{Z} _{p}(\mathcal{L}_{+};q, t)-\check{\mathcal{Z} _{p}(\mathcal{L}_{-};q, t)\equiv(-1)^{p-1}\check{\mathcal{Z} _{p}(\mathcal{L}_{0};q, t) \mathrm{m}\mathrm{o}\mathrm{d} \{p\}^{2} type I \check{\mathcal{Z} _{p}(\mathcal{L}_{+};q, t)-\check{\mathcal{Z} _{p}(\mathcal{L}_{-};q, t)\equiv(-1)^{p-1}p\{p\}^{2}\check{\mathcal{Z} _{p}(\mathcal{L}_{0};q, t) \mathrm{m}\mathrm{o}\mathrm{d} \{p\}^{2}\lceil p]^{2} \{p\}. where. =. q^{p} -q^{-p}, \lceil p ]. =. \mathbb{Z}[(q-q^{-1})^{2}, t^{\pm 1}].. \{p\}/\{1\}. .. The notation A. \equiv. B. \mathrm{m}\mathrm{o}\mathrm{d} C. type II. means. \displaystyle \frac{A-B}{c}. \in. examples are confirmed in [7]. excitingly, such congruence skein relations is not an isolated phenomenon. Chen‐ [12] first proved the integrality of composite invariants of full colored HOMFLY‐PT. Different kinds of More. Zhu. invariants and discovered. a. type II. congruence skein relation for them. But it looks like. skein relations for Kauffman invariants. This could. explain. the. mysterious mathematical. LMOV type conjecture proposed by Marino in [42], connecting the full colored HOMFLY‐PT invariants and the colored Kauffman invariants. We will study. nature of. an. congruence skein relations. has been obtained 3.4. A. new. of. the colored. Kauffman. invariants and substantial evidence. [9].. vision. on. proposing Volume Conjectures inspired by Congruence cyclotomic expansion. relations and Habiro type Habiro. [25]. plications. following cyclotomic expansion formula TQFT.. established the. in the. area. of. which has many ap‐.
(10) 65. Theorem 3.3. (Cyclotomic expansion. for colored Jones ,. J_{N}(\displaystyle\mathcal{K};q)=\sum_{k=0}^{N}C_{N+1},{ _{k}H_{k}(\mathcal{K}) where. (k-1. Habiro. polynomial,. H_{k}(\mathcal{K})\in \mathbb{Z}[q, q^{-1}] independent of N (N\geq 0). any knot \mathcal{K} , there exists. [25]).. For. such that. ,. ,. C_{N+1,k}=\{N-(k-1)\}\{N-(k-2)\}\cdots\{N-1\}\{N\}\{N+2\}\{N+3\}\cdots\{N+2+ for k=1, N and C_{N+1,0}=1 In particular, J_{0}(\mathcal{K};q)=H_{0}(\mathcal{K})=1. .. ,. We discover that the idea of. gap equations in cyclotomic expansions plays an impor‐ Conjectures. The root of unity used in original Volume Conjecture of gap equations \{N+1\}=0 where \{N+1\} serves as a gap in. tant role in Volume. is. \displayst le\frac{$\pi$\sqrt{-1}{N+1}. a. ,. solution. C_{N+1,k}.. Chen‐Liu‐Zhu. ,. [8]. discovered the. cyclotomic expansion of colored SU(n). invariants. were. indicated from their congruence relations. By studying the so called gap equation in cyclotomic expansions, we proposed a Volume Conjecture for the colored SU(n) invariants.. machinery will open a window to understand the very mysterious invariants, especially between Volume Conjectures and Habiro expansion type cyclotomic In [7], we proposed a congruence relation conjecture for the colored SU(n) invariants J_{N}^{SU(n)}(\mathcal{K};q) which is actually a consequence of the following conjecture. We think this. new. of quantum. essence. Conjecture. [8]).. 3.4. (Cyclotomic expansions. For any knot \mathcal{K} , there exist. that. H_{k}^{(n)}(\mathcal{K}). for colored. \mathbb{Z}[q, q^{-1}]. \in. ,. SU(n) invariants, independent of. J_{N}^{SU(n)}(\displaystyle\mathcal{K};q)=\sum_{k=0}^{N}C_{N+1}^{(n)},{ _{k}H_{k}^{(n)}(\mathcal{K}). where. N. Chen‐Liu‐Zhu. (N \geq 0). ,. such. ,. C_{N+1,k}^{(n)}=\{N-(k-1)\}\{N-(k-2)\}\cdots\{N-1\}\{N\}\{N+n\}\{N+n+1\}\cdots\{N+ for k=1, N and C_{N+1,0}^{(n)}=1 In particular, J_{0}^{SU(n)}(\mathcal{K};q)=H_{0}^{(n)}(\mathcal{K})=1.. n+(k-1. .. ,. We choose solutions of. of. gap equations as our roots of unity. In the above conjecture we could see that the gap equations associated to the basis expansions, cyclotomic of the cyclotomic expansion are \{N+a\}=0 for We introduce. C_{N+1,k}^{(n)}. roots of. unity. Then for. a. $\xi$_{N,a}(s)=\displaystyle \exp(\frac{s $\pi$\sqrt{-1} {N+a}). fixed n\geq 2 ,. we. a\in\{1, 2, n-1\}. ,. where a, s\in \mathbb{Z} , which. 3.5. these. gap equations. have. (Volume Conjecture a\in\{1, 2, n-1\} then Conjecture. satisfy. .. for colored. SU(n) invariants,. Chen‐Liu‐Zhu. ,. 2 $\pi$ s\displaystyle \lim_{N\rightar ow\infty}\frac{\log J_{N}^{SU(n)}(\mathcal{K};$\xi$_{N,a}(s) }{N+1}=Vol (S^{3}\backslash \mathcal{K})+\sqrt{-1}CS(S^{3}\backslash \mathcal{K}) for. any. hyperbolic. knot \mathcal{K}.. [8]). If.
(11) 66. 3.5. Applying. this. new. vision to. M. Khovanov introduced the idea of. Superpolynomials that the reduced Poincare. categorification. poly‐ homology \mathcal{P}(\mathcal{K};q, t) polynomial J(\mathcal{K};q) i.e. \mathcal{P}(\mathcal{K};q, -1) J(\mathcal{K};q) Then Khovanov‐Rozansky [32] generalized the categorification of the Jones polynomial to the categorification of the sl(N) invariants, whose corre‐ sponding Poincare polynomial \mathcal{P}^{sl(N)}(\mathcal{K};q, t) recovers the classical HOMFLY‐PT poly‐ nomial with specialization a q^{N} i.e. \mathcal{P}^{sl(N)}(\mathcal{K};q, -1) P(\mathcal{K};q^{N}, q) The idea of the superpolynomial \mathcal{P}(\mathcal{K};a, q, t) was introduced in [16] by Dunfield‐Gukov‐Rasmussen so that they could recover the classical HOMFLY‐PT polynomial and Alexander polyno‐ mial respectively. This was further studied by Khovanov‐Rozansky in [33]. The theory of the superpolynomial become a very active area which attracts many mathematician and physicists. Dunfield‐Gukov‐Rasmussen further argued [16] that the superpolynomial \mathcal{P}(\mathcal{K};a, q, t) could recover \mathcal{P}^{sl(N)}(\mathcal{K};q, t) under a certain differential d_{N} while the special‐ ized superpolynomial \mathcal{P}(\mathcal{K};t^{-1}, q, t) could recover the Poincare polynomial HFK(\mathcal{K};q^{2}, t) of Heegaard‐Floer knot homology \overline{HFK}_{i}(\mathcal{K};s) under a certain differential d_{0}. The author first proposed a congruence relation conjecture under some homological t‐ grading shifting just like the non‐categorified colored SU(n) invariants[7]. Based on these congruence relations, finally the author formulated the following cyclotomic expansion conjecture, nomial of Khovanovs =. recovers. .. =. =. ,. .. (Cyclotomic Expansion Conjecture of the Superpolynomial of colored homology, Chen [5]). For any knot \mathcal{K} there exists an integer‐valued invari‐ $\alpha$(\mathcal{K}) \in \mathbb{Z}, s.t the reduced Superpolynomial \mathcal{P}_{N}(\mathcal{K};a, q, t) of the colored HOMFLY‐PT. Conjecture. 3.6. HOMFLY‐PT ant. the Jones. ,. .. homology of. a. knot \mathcal{K} has the. following cyclotomic expansion formula. (-t)^{N $\alpha$(\mathcal{K})}\mathcal{P}_{N}(\mathcal{K};a, q, t). = 1+\displaystyle \sum_{k=1}^{N}H_{k}(\mathcal{K};a, q, t) (A_{-1}(a, q, t)\prod_{i=1}^{k}(\frac{\{N+1-i\} {\{i\} B_{N+i-1}(a, q, t) ) with coefficient functions H_{k}(\mathcal{K};a, q, t) \in \mathbb{Z}[a^{\pm 1}, q^{\pm 1}, t^{\pm 1}] B_{i}(a, q, t)=t^{2}aq^{i}+t^{-1}a^{-1}q^{-i} and \{p\}=q^{p}-q^{-p}.. Remark 3.1. The above. ,. where. A_{i}(a, q, t)=aq^{i}+t^{-1}a^{-1}q^{-i},. Conjecture‐Definition for the invariant $\alpha$(\mathcal{K}) should be understood conjecture of a knot \mathcal{K} is true for N=1 then $\alpha$(\mathcal{K}) is defined.. in this way. If the above. ,. We tested many homologically thick knots up to 10 crossings to illustrate this con‐ jecture as well as many examples with higher representation. Based on highly nontrivial. computations of. torus. knots/links. studied in. theorem for torus knots. Theorem 3.7. (Chen [5]).. expansion conjecture. For any. is true. 1) (n-1)/2. Now. we are. considering. a. for. [17],. we. are. able to prove the. coprime pair (m, n)=1 where. torus knot. ,. T(m, n). problem relating. and. we. have. to the sliceness of. a. m<n ,. the. $\alpha$(T(m, n)) knot.. following. cyclotomic =. -(m-.
(12) 67. Definition 3.1. The smooth 4‐ball genus g_{4}(\mathcal{K}) of a knot \mathcal{K} is the minimum genus of a surface smoothly embedded in the 4‐ball B^{4} with boundary the knot. In particular, \mathrm{a} knot. \mathcal{K}\subset S^{3} is called smoothly slice if g_{4}(\mathcal{K})=0.. Conjecture 3.8 (Milnor Conjecture, proved by Kronheimer‐Mrowka The smooth 4‐ball genus for torus knot T(m, n) is (m-1)(n-1)/2. Based. all the above. on. results,. Conjecture 3.9 (Chen [5]). The conjecture for N= 1 ) is a lower. g_{4}(\mathcal{K}). we are. able to propose the. invariant. bound. for. following conjecture.. $\alpha$(\mathcal{K}) (determined by the smooth. 4‐ball. Rasmussen).. and. the. cyclotomic expansion. g_{4}(\mathcal{K}). genus. ,. Remark 3.2. Rasmussen. [55]. introduced. a. knot concordant invariant. bound for the smooth 4‐ball genus for knots. For all the knots to the Ozsváth‐Szabós $\tau$ invariant and Rasmussens s invariant. we. s(\mathcal{K}). ,. which is. \leq. tested,. (up. to. a. we. present certain motivation. to propose. our. Volume. factor of. and. lower. 2).. for reduced. \displaystyle \mathcal{P}_{N}(4_{1};a, q, t) = 1+ \sum_{k=1}^{N}\prod_{i=1}^{k}(\frac{\{N+1-i\} {\{i\} A_{i-2}(a, q, t)B_{N-1+i}(a, q, t) A_{i}(a, q, t)=aq^{i}+t^{-1}a^{-1}q^{-i}, B_{i}(a, q, t)=t^{2}aq^{i}+t^{-1}a^{-1}q^{-i}. a. it is identical. Conjecture superpolynomials associated to the colored HOMFLY‐PT homologies. We have the following expression for the figure‐eight knot 4_{1} [27], Now. where. | $\alpha$(\mathcal{K})|. i.e.. .. ,. \{p\}=q^{p}-q^{-p}.. apply the idea of gap equations on the cyclotomic expansion of reduced superpolynomial of the colored HOMFLY‐PT homology. By looking at middle terms Now. we. A_{N-2}(q^{n}, q, t)=q^{N+n-2}+t^{-1}q^{-(N+n-2)}. to know that the. and. B_{N}(q^{n}, q, t)=(-t)^{2}q^{N+n}+t^{-1}q^{-(N+n)} we get equation (-t)q^{N+n-1}+t^{-1}q^{-(N+n-1)}=0. ,. is the. possible gap equations By studying the above 2‐variable gap equations, we propose a Volume Conjecture for superpolynomials of the HOMFLY‐PT homology at its solution t=q^{-(N+n-1)} as follows.. Conjecture 3.10 (Volume Conjecture for the Superpolynomial of homology, Chen [5]). For any hyperbolic knot \mathcal{K} we have. the HOMFLY‐PT. ,. 2$\pi$\displaystyle\lim_{N\rightar ow\infty}\frac{\log\mathcal{P}_{N}(\mathcal{K};q^{n},q ^{-(N+n-1)} |_{q=e^{\frac{$\pi$\sqrt{-1} {N+b} {N+1}=Vol where b\geq 1 and. \displaystyle \frac{n-1-b}{2}. is not. a. (S^{3}\backslash \mathcal{K})+\sqrt{-1}CS(S^{3}\backslash \mathcal{K}) (mod\sqrt{-1}$\pi$^{2}\mathbb{Z}). positive integer.. Remark 3.3. The choices of roots of unity in this conjecture are much more relaxed than original Volume conjectures, because here b can be any large positive integers.. those of the For. example, the original Volume Conjecture is only proposed for n=2 Conjecture is proposed for all positive integer b with n=2.. and b= 1 , but. this Volume. interesting to know the relationship of this volume conjecture to the one proposed [19], where they used categorified \mathrm{A} ‐polynomials of knots. We think this new vision on Volume Conjecture creates many problems, which opens a new window to understand the very mysterious essence of quantum invariants, and form another key goal of this survey. It will be in. ,.
(13) 68. Cyclotomic expansion for the reduced superpolynomial \mathcal{F}_{N}(\mathcal{K};a, q, t) of the colored homology (Gukov‐Walcher [24]) were also proposed by the author in [5]. The author address the following question in Problem set of ITLDT conference. From the above discussion and many results we have obtained, the author have a feeling that all quantum invariants may have such cyclotomic expansion and will also have Volume Conjecture from studying of such Gap equations indicated from the corresponding cyclotomic expansion. Kauffman. Here is. a. summary of various results.. Relations between various HOMFLY‐PT theories.
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(18) 73. Department of Mathematics,. ETH Zurich. 8092 Zurich. Switzerland \mathrm{E} ‐mail address:. qingtao.. [email protected].
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