Abstract
So far the representation theory behind the skein, Jones, and Alexander polynomial was not used. We will give some applications of it now. The theory is well explained in [J]. We will use Jones’ conventions, unless otherwise specified.
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- 1.
Here consistently “tableau” is used as a synonym for (Young) diagram, i.e. with no additional information attached to it.
- 2.
In the parametrization P(v, z), used by Morton–Short, unlike for X, the coefficients of \(P(\hat{\beta }_{k,l})\) become quickly large and produce machine size integer overflows. In particular, we could not calculate correctly polynomials for | l | > 2.
- 3.
This is not to be confused with the variable e, which we use for exponent sum of a braid, or with the idempotent e i from Jones’ lemma 9.1.
- 4.
The second duplication was noted in the remarks after Jones’ table, but not referred to correctly in its last column.
References
J.W. Alexander, A lemma on systems of knotted curves. Proc. Natl. Acad. Sci. USA 9, 93–95 (1923)
J.W. Alexander, Topological invariants of knots and links. Trans. Am. Math. Soc. 30, 275–306 (1928)
E. Artin, Theorie der Zöpfe, Abh. Math. Sem. Hamburgischen Univ. 4, 47–72 (1926)
E. Artin, Theory of braids. Ann. Math. 48(2), 101–126 (1947)
J.S. Birman, On the Jones polynomial of closed 3-braids. Invent. Math. 81(2), 287–294 (1985)
D. Bennequin, Entrelacements et équations de Pfaff. Soc. Math. de France, Astérisque 107–108, 87–161 (1983)
S. Bigelow, Representations of braid groups, in Proceedings of the International Congress of Mathematicians, vol. II (2002), pp. 37–45
J.S. Birman, K. Ko, S.J. Lee, A new approach to the word and conjugacy problems in the braid groups. Adv. Math. 139(2), 322–353 (1998)
S.A. Bleiler, Realizing concordant polynomials with prime knots. Pac. J. Math. 100(2), 249–257 (1982)
R.D. Brandt, W.B.R. Lickorish, K. Millett, A polynomial invariant for unoriented knots and links. Invent. Math. 74, 563–573 (1986)
J.S. Birman, W.W. Menasco, Studying knots via braids III: classifying knots which are closed 3 braids. Pac. J. Math. 161, 25–113 (1993)
J.S. Birman, W.W. Menasco, Studying links via closed braids II: On a theorem of Bennequin. Topol. Appl. 40(1), 71–82 (1991)
J.S. Birman, W.W. Menasco, Studying knots via braids VI: a non-finiteness theorem. Pac. J. Math. 156, 265–285 (1992)
J.S. Birman, H. Wenzl, Braids, link polynomials and a new algebra. Trans. Am. Math. Soc. 313(1), 249–273 (1989)
A. Champanerkar, I. Kofman, On the Mahler measure of Jones polynomials under twisting. Algebr. Geom. Topol. 5, 1–22 (2005)
P.R. Cromwell, Homogeneous links. J. Lond. Math. Soc. (series 2) 39, 535–552 (1989)
O. Dasbach, X.-S. Lin, On the head and the tail of the colored Jones polynomial. Compos. Math. 142(5), 1332–1342 (2006)
S. Eliahou, L.H. Kauffman, M. Thistlethwaite, Infinite families of links with trivial Jones polynomial. Topology 42(1), 155–169 (2003)
T. Fiedler, A small state sum for knots. Topology 32(2), 281–294 (1993)
T. Fiedler, Gauss Sum Invariants for Knots and Links. Mathematics and Its Applications, vol. 532 (Kluwer Academic Publishers, Boston, 2001)
T. Fiedler, V. Kurlin, A one-parameter approach to links in a solid torus. J. Math. Soc. Jpn. 62(1), 167–211 (2010). math.GT/0606381
H. Fujii, Geometric indices and the Alexander polynomial of a knot. Proc. Am. Math. Soc. 124(9), 2923–2933 (1996)
J. Franks, R.F. Williams, Braids and the Jones-Conway polynomial. Trans. Am. Math. Soc. 303, 97–108 (1987)
P. Freyd, J. Hoste, W.B.R. Lickorish, K. Millett, A. Ocneanu, D. Yetter, A new polynomial invariant of knots and links. Bull. Am. Math. Soc. 12, 239–246 (1985)
D. Gabai, The Murasugi sum is a natural geometric operation, in Low-Dimensional Topology (San Francisco, California, 1981). Contemporary Mathematics, vol. 20 (American Mathematical Society, Providence, RI, 1983), pp. 131–143
D. Gabai, The Murasugi sum is a natural geometric operation II, in Combinatorial Methods in Topology and Algebraic Geometry (Rochester, N.Y., 1982). Contemporary Mathematics, vol. 44 (American Mathematical Society, Providence, RI, 1985), pp. 93–100
D. Gabai, Detecting fibred links in S 3. Comment. Math. Helv. 61(4), 519–555 (1986)
F. Garside, The braid group and other groups. Q. J. Math. Oxford 20, 235–264 (1969)
M. Hirasawa, K. Murasugi, Double-torus fibered knots and pre-fiber surfaces. Musubime to Teijigen Topology (Dec. 1999), pp. 43–49
J. Hoste, M. Thistlethwaite, KnotScape, a knot polynomial calculation and table access program. Available at http://www.math.utk.edu/~morwen
J. Hoste, M. Thistlethwaite, J. Weeks, The first 1,701,936 knots. Math. Intell. 20(4), 33–48 (1998)
M. Ishikawa, On the Thurston-Bennequin invariant of graph divide links. Math. Proc. Camb. Philos. Soc. 139(3), 487–495 (2005)
V.F.R. Jones, Hecke algebra representations of braid groups and link polynomials. Ann. Math. 126, 335–388 (1987)
V.F.R. Jones, A polynomial invariant of knots and links via von Neumann algebras. Bull. Am. Math. Soc. 12, 103–111 (1985)
T. Kanenobu, Relations between the Jones and Q polynomials of 2-bridge and 3-braid links, Math. Ann. 285, 115–124 (1989)
T. Kanenobu, Examples on polynomial invariants of knots and links II. Osaka J. Math. 26(3), 465–482 (1989)
T. Kanenobu, Examples on polynomial invariants of knots and links. Math. Ann. 275, 555–572 (1986)
T. Kanenobu, An evaluation of the first derivative of the Q polynomial of a link. Kobe J. Math. 5(2), 179–184 (1988)
L.H. Kauffman, An invariant of regular isotopy. Trans. Am. Math. Soc. 318, 417–471 (1990)
L.H. Kauffman, State models and the Jones polynomial. Topology 26, 395–407 (1987)
M.E. Kidwell, On the degree of the Brandt-Lickorish-Millett-Ho polynomial of a link. Proc. Am. Math. Soc. 100(4), 755–762 (1987)
O. Kakimizu, Classification of the incompressible spanning surfaces for prime knots of ≤ 10 crossings. Hiroshima Math. J. 35, 47–92 (2005)
J.A. Kneissler, Woven braids and their closures. J. Knot Theory Ramifications 8(2), 201–214 (1999)
T. Kobayashi, Uniqueness of minimal genus Seifert surfaces for links. Topol. Appl. 33(3), 265–279 (1989)
D. Kreimer, Knots and Feynman Diagrams. Cambridge Lecture Notes in Physics, vol. 13 (Cambridge University Press, Cambridge, 2000)
K. Kawamuro, The algebraic crossing number and the braid index of knots and links. Algebr. Geom. Topol. 6, 2313–2350 (2006)
W.B.R. Lickorish, K.C. Millett, A polynomial invariant for oriented links. Topology 26(1), 107–141 (1987)
W.B.R. Lickorish, M.B. Thistlethwaite, Some links with non-trivial polynomials and their crossing numbers. Comment. Math. Helv. 63, 527–539 (1988)
H.R. Morton, Seifert circles and knot polynomials. Proc. Camb. Philos. Soc. 99, 107–109 (1986)
H.R. Morton, (ed.), Problems, in Braids, Santa Cruz, 1986, ed. by J.S. Birman, A.L. Libgober. Contemporary Mathematics, vol. 78 (Cambridge University Press, Cambridge, 1986), pp. 557–574
K. Murasugi, J. Przytycki, The skein polynomial of a planar star product of two links. Math. Proc. Camb. Philos. Soc. 106(2), 273–276 (1989)
K. Murasugi, J. Przytycki, An index of a graph with applications to knot theory. Mem. Am. Math. Soc. 106(508) (1993)
J. Murakami, The Kauffman polynomial of links and representation theory. Osaka J. Math. 24(4), 745–758 (1987)
H.R. Morton, H.B. Short, The2-variable polynomial of cable knots. Math. Proc. Camb. Philos. Soc. 101(2), 267–278 (1987)
H.R. Morton, H.B. Short, br9z.p, a Pascal program for calculation of the skein polynomial from braids. http://www.liv.ac.uk/~su14/knotprogs.html
W.W. Menasco, M.B. Thistlethwaite, The Tait flyping conjecture. Bull. Am. Math. Soc. 25(2), 403–412 (1991)
J. Malesic, P. Traczyk, Seifert circles, braid index and the algebraic crossing number. Topol. Appl. 153(2–3), 303–317 (2005)
K. Murasugi, On the braid index of alternating links. Trans. Am. Math. Soc. 326(1), 237–260 (1991)
K. Murasugi, On Closed 3-Braids. Memoirs AMS, vol. 151 (American Mathematical Society, Providence, RI, 1974)
K. Murasugi, Jones polynomial and classical conjectures in knot theory. Topology 26, 187–194 (1987)
T. Nakamura, Notes on the braid index of closed positive braids. Topology Appl. 135(1–3), 13–31 (2004)
T. Nakamura, Braidzel surfaces and the Alexander polynomial, Proceedings of the Workshop “Intelligence of Low Dimensional Topology”, Osaka City University (2004), pp. 25–34
Y. Ni, Closed 3-braids are nearly fibred. J. Knot Theory Ramifications 18(12), 1637–1649 (2009). math.GT/0510243
Y. Ohyama, On the minimal crossing number and the braid index of links. Canad. J. Math. 45(1), 117–131 (1993)
S. Orevkov, Quasipositivity problem for 3-braids. Turk. J. Math. 28, 89–93 (2004). Also available at https://www.math.univ-toulouse.fr/~orevkov/
M. Polyak, O. Viro, Gauss diagram formulas for Vassiliev invariants. Int. Math. Res. Notes 11, 445–454 (1994)
M. Polyak, O. Viro, On the Casson knot invariant. J. Knot Theory Ramifications 10(5), 711–738 (2001). Knots in Hellas ’98, vol. 3 (Delphi)
D. Rolfsen, Knots and Links (Publish or Perish, New York, 1976)
L. Rudolph, Braided surfaces and Seifert ribbons for closed braids. Comment. Math. Helv. 58, 1–37 (1983)
L. Rudolph, Quasipositivity as an obstruction to sliceness. Bull. Am. Math. Soc. (N.S.) 29(1), 51–59 (1993)
O. Schreier, Über die Gruppen A a B b = 1, Abh. Math. Sem. Univ. Hamburg 3, 167–169 (1924)
C. Squier, The Burau representation is unitary. Proc. Am. Math. Soc. 90, 199–202 (1984)
D. Silver, A. Stoimenow, S.G. Williams, Euclidean Mahler measure and twisted links. Algebr. Geom. Topol. 6, 581–602 (2006). math.GT/0412513
C. Sundberg, M.B. Thistlethwaite, The rate of growth of the number of prime alternating links and tangles. Pac. J. Math. 182(2), 329–358 (1998)
A. Stoimenow, On polynomials and surfaces of variously positive links. J. Eur. Math. Soc. 7(4), 477–509 (2005). math.GT/0202226
A. Stoimenow, The skein polynomial of closed 3-braids. J. Reine Angew. Math. 564, 167–180 (2003)
A. Stoimenow, On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks. Trans. Am. Math. Soc. 354(10), 3927–3954 (2002)
A. Stoimenow, Coefficients and non-triviality of the Jones polynomial, J. Reine Angew. Math. 657 (2011), 1–55; see also math.GT/0606255
A. Stoimenow, Diagram Genus, Generators and Applications. Monographs and Research Notes in Mathematics (T&F/CRC Press, Boca Raton, 2016). ISBN 9781498733809
A. Stoimenow, The braid index and the growth of Vassiliev invariants. J. Knot Theory Ram. 8(6), 799–813 (1999)
A. Stoimenow, Positive knots, closed braids, and the Jones polynomial. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2(2), 237–285 (2003). math.GT/9805078
A. Stoimenow, Knots of (canonical) genus two. Fund. Math. 200(1), 1–67 (2008). math.GT/0303012
A. Stoimenow, A. Vdovina, Counting alternating knots by genus. Math. Ann. 333, 1–27 (2005)
D. Silver, S. Williams, Coloring link diagrams with a continuous palette. Topology 39, 1225–1237 (2000)
M. B. Thistlethwaite, On the Kauffman polynomial of an adequate link. Invent. Math. 93(2), 285–296 (1988)
M.B. Thistlethwaite, A spanning tree expansion for the Jones polynomial. Topology 26, 297–309 (1987)
P. Traczyk, 3-braids with proportional Jones polynomials. Kobe J. Math. 15(2), 187–190 (1998)
P. Vogel, Representation of links by braids: a new algorithm. Comment. Math. Helv. 65, 104–113 (1990)
R.F. Williams, Lorenz knots are prime. Ergodic Theory Dyn. Syst. 4(1), 147–163 (1984)
S. Wolfram, Mathematica — A System for Doing Mathematics by Computer (Addison-Wesley, Reading, 1989)
P. Xu, The genus of closed 3-braids. J. Knot Theory Ramifications 1(3), 303–326 (1992)
S. Yamada, The minimal number of Seifert circles equals the braid index. Invent. Math. 88, 347–356 (1987)
Y. Yokota, Polynomial invariants of positive links. Topology 31(4), 805–811 (1992)
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Stoimenow, A. (2017). Applications of the Representation Theory. In: Properties of Closed 3-Braids and Braid Representations of Links . SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-68149-8_7
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