Geodesic
Are there $p$-adic knot invariants?
Teoretičeskaâ i matematičeskaâ fizika, Tome 187 (2016) no. 1, pp. 3-11 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We suggest using the Hall–Littlewood version of the Rosso–Jones formula to define the germs of $p$-adic HOMFLY-PT polynomials for torus knots $[m,n]$ as coefficients of superpolynomials in a $q$-expansion. In this form, they have at least the $[m,n]\leftrightarrow[n,m]$ topological invariance. This opens a new possibility to interpret superpolynomials as $p$-adic deformations of HOMFLY polynomials and poses a question of generalizing to other knot families, which is a substantial problem for several branches of modern theory.
Keywords: knot polynomial, $p$-adic analysis, $p$-adic string. 
@article{TMF_2016_187_1_a0,
     author = {A. Yu. Morozov},
     title = {Are there $p$-adic knot invariants?},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {3--11},
     year = {2016},
     volume = {187},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2016_187_1_a0/}
}
TY  - JOUR
AU  - A. Yu. Morozov
TI  - Are there $p$-adic knot invariants?
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2016
SP  - 3
EP  - 11
VL  - 187
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2016_187_1_a0/
LA  - ru
ID  - TMF_2016_187_1_a0
ER  - 
%0 Journal Article
%A A. Yu. Morozov
%T Are there $p$-adic knot invariants?
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2016
%P 3-11
%V 187
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2016_187_1_a0/
%G ru
%F TMF_2016_187_1_a0
A. Yu. Morozov. Are there $p$-adic knot invariants?. Teoretičeskaâ i matematičeskaâ fizika, Tome 187 (2016) no. 1, pp. 3-11. http://geodesic.mathdoc.fr/item/TMF_2016_187_1_a0/

[1] I. V. Volovich, Class. Quantum Gravity, 4:4 (1987), L83–L87 ; Yu. I. Manin, “Reflections on arithmetical physics”, Conformal Invariance and String Theory (Poiana Braşov, 1987), Academic Press, Boston, MA, 1989, 293–303 ; $p$-Adic Numbers Ultrametric Anal. Appl., 5:4 (2013), 313–325, arXiv: ; I. Ya. Aref'eva, B. G. Dragović, I. V. Volovich, Phys. Lett. B, 200:4 (1988), 512–514 ; 209:4 (1988), 445–450 ; 212:3 (1988), 283–291 ; 214:3 (1988), 339–349 ; A. V. Zabrodin, Commun. Math. Phys., 123:3 (1989), 463–483 ; A. Levin, A. Morozov, Phys. Lett. B, 243:3 (1990), 207–214 ; B. Dragović, Phys. Lett. B, 256:3–4 (1991), 392–396 ; J. Math. Phys., 34:3 (1992), 1143–1148, arXiv: ; Б. Г. Драгович, ТМФ, 93:2 (1992), 211–218 ; 100:3 (1994), 342–353 ; L. Brekke, P. G. O. Freund, Phys. Rep., 233:1 (1993), 1–66 ; V. Vladimirov, I. Volovich, E. Zelenov, $p$-Adic Analysis and Mathematical Physics, Series on Soviet and East European Mathematics, 1, World Sci., Singapore, 1994 ; G. Djordjević, B. Dragovich, L. Nešić, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 6:2 (2003), 179–195, arXiv: ; Modern Phys. Lett. A, 14:5 (1999), 317–325, arXiv: ; B. Dragovich, A. Yu Khrennikov, S. V. Kozyrev, I. V. Volovich, $p$-Adic Numbers Ultrametric Anal. Appl., 1:1 (2009), 1–17, arXiv: 1312.5160math-ph/0402037hep-th/0105030hep-th/00052160904.4205 | DOI | MR | DOI | MR | DOI | MR | Zbl | DOI | MR | DOI | MR | DOI | MR | DOI | MR | DOI | MR | Zbl | DOI | MR | DOI | MR | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | MR | DOI | MR | Zbl | DOI | MR | DOI | MR | Zbl

[2] L. Brekke, P. G. O. Freund, M. Olson, E. Witten, Nucl. Phys. B, 302:3 (1988), 365–402 | DOI | MR

[3] L. O. Chekhov, A. D. Mironov, A. V. Zabrodin, Commun. Math. Phys., 125:4 (1989), 675–711 ; Д. Р. Лебедев, А. Ю. Морозов, ТМФ, 82:1 (1990), 3–10 | DOI | MR | Zbl | MR

[4] J. W. Alexander, Trans. Amer. Math. Soc., 30:2 (1928) ; J. H. Conway, “An enumeration of knots and links, and some of their algebraic properties”, Computational Problems in Abstract Algebra (Science Research Council Atlas Computer Laboratory, Oxford, 29 August – 2 September, 1967), ed. J. Leech, Pergamon Press, Oxford, New York, 1970, 329–358 ; V. F. R. Jones, Invent. Math., 72:1 (1983), 1–25 ; Bull. Amer. Math. Soc. (N. S.), 12:1 (1985), 103–111 ; Ann. Math., 126:2 (1987), 335–388 ; L. H. Kauffman, Topology, 26:3 (1987), 395–407 ; P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, A. Ocneanu, Bull. Amer. Math. Soc. (N. S.), 12:2 (1985), 239–246 ; J. H. Przytycki, K. P. Traczyk, Kobe J. Math., 4:2 (1987), 115–139 | DOI | MR | Zbl | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | MR | Zbl

[5] A. Mironov, A. Morozov, And. Morozov, JHEP, 03 (2012), 034, 33 pp., arXiv: ; S. Nawata, P. Ramadevi, Zodinmawia, J. Knot Theory Ramifications, 22:13 (2013), 1350078, 58 pp., arXiv: ; А. С. Анохина, А. А. Морозов, ТМФ, 178:1 (2014), 3–68, arXiv: ; A. Mironov, A. Morozov, Nucl. Phys. B, 899 (2015), 395–413, arXiv: ; A. Mironov, A. Morozov, An. Morozov, A. Sleptsov, Colored knot polynomials. HOMFLY in representation $[2,1]$, Internat. J. Modern Phys. A, 30, no. 26, 2015, 40 pp., arXiv: 1112.26541302.51441307.22161506.003391508.02870 | DOI | MR | DOI | MR | Zbl | DOI | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[6] Knot Polynomials and Other Link Invariants, (2015) http://knotebook.org

[7] S.-S. Chern, J. Simons, Proc. Nat. Acad. Sci. U.S.A., 68:4 (1971), 791–794 ; Ann. Math. (2), 99:1 (1974), 48–69 ; A. S. Schwarz, “New topological invariants arising in the theory of quantized fields”, Topology and Its Applications, Talk at the International Topological Conference. Abstracts. Part 2 (Baku, October 3–8, 1987), AMS, Providence, RI, 1993 ; M. Atiyah, The Geometry and Physics of Knots, Cambridge Univ. Press, Cambridge, 1990 | DOI | MR | Zbl | DOI | MR | Zbl | Zbl | MR

[8] E. Witten, Commun. Math. Phys., 121:3 (1989), 351–399 | DOI | MR | Zbl

[9] A. Morozov, A. Smirnov, Nucl. Phys. B, 835:3 (2010), 284–313, arXiv: ; A. Smirnov, Notes on Chern–Simons theory in the temporal gauge, arXiv: 1001.20030910.5011 | DOI | MR | Zbl

[10] R. K. Kaul, T. R. Govindarajan, Nucl. Phys. B, 380:1–2 (1992), 293–333, arXiv: ; 393:1–2 (1993), 392–412 ; P. Rama Devi, T. R. Govindarajan, R. K. Kaul, Nucl. Phys. B, 402:1–2 (1993), 548–566, arXiv: ; Nucl. Phys. B, 422:1–2 (1994), 291–306, arXiv: ; Modern Phys. Lett. A, 10:22 (1995), 1635–1658, arXiv: ; Zodinmawia, P. Ramadevi, $SU(N)$ quantum Racah coefficients and non-torus links, arXiv: ; Reformulated invariants for non-torus knots and links, arXiv: ; D. Galakhov, D. Melnikov, A. Mironov, A. Morozov, A. Sleptsov, Phys. Lett. B, 743 (2015), 71–74, arXiv: ; A. Mironov, A. Morozov, A. Sleptsov, JHEP, 07 (2015), 069, 34 pp., arXiv: ; D. Galakhov, D. Melnikov, A. Mironov, A. Morozov, A. Sleptsov, Nucl. Phys. B, 899 (2015), 194–228, arXiv: ; A. Mironov, A. Morozov, And. Morozov, P. Ramadevi, V. K. Singh, JHEP, 07 (2015), 109, 68 pp., arXiv: hep-th/9111063hep-th/9212110hep-th/9312215hep-th/94120841107.39181209.13461412.26161412.84321502.026211504.00371 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | MR | DOI | MR | Zbl | DOI | DOI | MR | Zbl | DOI

[11] A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, Nucl. Phys. B, 241:2 (1984), 333–380 ; А. Б. Замолодчиков, Ал. Б. Замолодчиков, Конформная теория поля и критические явления в двумерных системах, МЦНМО, М., 2009 ; А. Д. Миронов, С. А. Миронов, А. Ю. Морозов, А. А. Морозов, ТМФ, 165:3 (2010), 503–542, arXiv: 0908.2064 | DOI | MR | Zbl | MR | DOI | DOI | Zbl

[12] E. Witten, Commun. Math. Phys., 92:4 (1984), 455–472 | DOI | MR | Zbl

[13] M. Wakimoto, Commun. Math. Phys., 104:4 (1986), 605–609 ; А. Морозов, Письма в ЖЭТФ, 49:6 (1989), 305–308 ; A. Gerasimov, A. Morozov, M. Olshanetsky, A. Marshakov, S. Shatashvili, Internat. J. Modern Phys. A, 5:13 (1990), 2495–2589 | DOI | MR | Zbl | MR | DOI | MR

[14] P. G. O. Freund, A. V. Zabrodin, Commun. Math. Phys., 147:2 (1992), 277–294, arXiv: hep-th/9110066 | DOI | MR | Zbl

[15] H. Itoyama, A. Mironov, A. Morozov, And. Morozov, Internat. J. Modern Phys. A, 28:3–4 (2013), 1340009, 81 pp., arXiv: 1209.6304 | DOI | MR | Zbl

[16] S. Garoufalidis, T. T. Q. Lê, Geom. Topol., 9 (2005), 1253–1293, arXiv: ; An analytic version of the Melvin–Morton–Rozansky conjecture, arXiv: ; A. Mironov, A. Morozov, AIP Conf. Proc., 1483 (2012), 189–211, arXiv: ; S. Garoufalidis, P. Kucharski, P. Sułkowski, Knots, BPS states, and algebraic curves, arXiv: math/0309214math/05036411208.22821504.06327 | DOI | MR | Zbl | DOI

[17] A. Alexandrov, A. Mironov, A. Morozov, Internat. J. Modern Phys. A, 19:24 (2004), 4127–4165, arXiv: ; 21:12 (2006), 2481–2518, arXiv: ; Fortsch. Phys., 53:5–6 (2005), 512–521, arXiv: ; Physica D, 235:1–2 (2007), 126–167, arXiv: ; JHEP, 12 (2009), 053, 49 pp., arXiv: ; А. С. Александров, А. Д. Миронов, А. Ю. Морозов, ТМФ, 150:2 (2007), 179–192, arXiv: ; L. Chekhov, B. Eynard, N. Orantin, JHEP, 12 (2006), 053, 31 pp., arXiv: ; B. Eynard, N. Orantin, Commun. Number Theory Phys., 1:2 (2007), 347–452, arXiv: ; R. Dijkgraaf, H. Fuji, M. Manabe, Nucl. Phys. B, 849:1 (2011), 166–211, arXiv: ; J. Gu, H. Jockers, A. Klemm, M. Soroush, Commun. Math. Phys., 336:2 (2015), 987–1051, arXiv: hep-th/0310113hep-th/0412099hep-th/0412205hep-th/06082280906.3305hep-th/0605171math-ph/0603003math-ph/07020451010.45421401.5095 | DOI | MR | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[18] P. Dunin-Barkowski, A. Mironov, A. Morozov, A. Sleptsov, A. Smirnov, JHEP, 03 (2013), 021, 85 pp., arXiv: 1106.4305 | DOI | MR

[19] M. Rosso, V. F. R. Jones, J. Knot Theory Ramifications, 2:1 (1993), 97–112 ; X.-S. Lin, H. Zheng, Trans. Amer. Math. Soc., 362:1 (2010), 1–18, arXiv: math/0601267 | DOI | MR | Zbl | DOI | MR | Zbl

[20] I. Makdonald, Simmetricheskie funktsii i mnogochleny Kholla, Mir, M., 1984 | MR | Zbl

[21] M. Khovanov, Duke Math. J., 101:3 (2000), 359–426, arXiv: ; S. Gukov, A. Schwarz, C. Vafa, Lett. Math. Phys., 74:1 (2005), 53–74, arXiv: ; M. Khovanov, L. Rozansky, Fund. Math., 199:1 (2008), 1–91, arXiv: ; Geom. Topol., 12:3 (2008), 1387–1425, arXiv: ; Virtual crossings, convolutions and a categorification of the $SO(2N)$ Kauffman polynomial, arXiv: ; V. Dolotin, A. Morozov, Nucl. Phys. B, 878 (2014), 12–81, arXiv: math/9908171hep-th/0412243math/0401268math/0505056math/07013331308.5759 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[22] A. Mironov, A. Morozov, S. Shakirov, A. Sleptsov, JHEP, 05 (2012), 70, 57 pp., arXiv: ; A. Mironov, A. Morozov, Sh. Shakirov, J. Phys. A: Math. Theor., 45:35 (2012), 355202, 11 pp., arXiv: ; P. Etingof, E. Gorsky, I. Losev, Adv. Math., 277 (2015), 124–180, arXiv: ; Sh. Shakirov, Colored knot amplitudes and Hall–Littlewood polynomials, arXiv: 1201.33391203.06671304.34121308.3838 | DOI | DOI | MR | Zbl | DOI | MR | Zbl

[23] F. I. Mautner, Amer. J. Math., 80 (1958), 441–457 ; P. Cartier, Harmonic Analysis on Homogeneous Spaces (William College, Williamstown, MA, July 31 – August 18, 1972), Proceedings of Symposia in Pure Mathematics, 26, AMS, Providence, RI, 1973 ; P. Freund, Phys. Lett. B, 257:1–2 (1991), 119–124 | DOI | MR | Zbl | MR | DOI | MR

[24] Ya. Kononov, A. Morozov, Pisma v ZhETF, 101:12 (2015), 931–934, arXiv: 1504.07146

[25] N. Dunfield, S. Gukov, J. Rasmussen, Experimental Math., 15:2 (2006), 129–159, arXiv: ; A. Mironov, A. Morozov, And. Morozov, AIP Conf. Proc., 1562 (2013), 123–155, arXiv: ; С. Б. Артамонов, А. Д. Миронов, А. Ю. Морозов, ТМФ, 179:2 (2014), 147–188, arXiv: math/05056621306.31971306.5682 | DOI | MR | Zbl | DOI | DOI

[26] A. Mironov, A. Morozov, And. Morozov, “Character expansion for HOMFLY polynomials I: Integrability and difference equations”, Strings, Gauge Fields, and the Geometry Behind (The Legacy of Maximilian Kreuzer), eds. A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov, E. Scheidegger, World Sci., Singapore, 2013, 101–118, arXiv: 1112.5754 | MR

[27] L. H. Kauffman, European J. Combin., 20:7 (1999), 663–690, arXiv: ; R. Fenn, D. P. Ilyutko, L. H. Kauffman, V. O. Manturov, “Unsolved problems in virtual knot theory and combinatorial knot theory”, Knots in Poland III. Part III, Banach Center Publications, 103, Polish Acad. Sci. Inst. Math., Warsaw, 2014, 9–61, arXiv: math/98110281409.2823 | DOI | MR | Zbl | DOI | MR

[28] N. Yu. Reshetikhin, V. G. Turaev, Commun. Math. Phys., 127:1 (1990), 1–26 | DOI | MR | Zbl

[29] A. Morozov, And. Morozov, Ant. Morozov, Phys.Lett. B, 737 (2014), 48–56, arXiv: ; L. Bishler, A. Morozov, An. Morozov, Ant. Morozov, Internat. J. Modern Phys. A, 30:14 (2015), 1550074, 39 pp., arXiv: ; A. Morozov, An. Morozov, A. Popolitov, Phys. Lett. B, 749 (2015), 309–325, arXiv: ; Matrix model and dimensions at hypercube vertices, arXiv: 1407.63191411.25691506.075161508.01957 | DOI | Zbl | DOI | MR | Zbl | DOI