Polynomial Knots

It is known that every knot in \(\mathbf{S}^3\) can be represented as the closure of the image of a polynomial embedding \(\mathbf{R} \to \mathbf{R}^3 \subset \mathbf{S}^3\).  See some examples there .

d9glass

Related publications

  • [PDF] [DOI] P. Koseleff, D. Pecker, F. Rouillier, and C. Tran, “Computing Chebyshev knot diagrams,” Journal of Symbolic Computation, vol. 86, p. 21, 2018.
    [Bibtex]
    @article{koseleff:hal-01232181,
    TITLE = {{Computing Chebyshev knot diagrams}},
    AUTHOR = {Koseleff, Pierre-Vincent and Pecker, Daniel and Rouillier, Fabrice and Tran, Cuong},
    URL = {https://hal.inria.fr/hal-01232181},
    JOURNAL = {{Journal of Symbolic Computation}},
    HAL_LOCAL_REFERENCE = {hal-01232181},
    PUBLISHER = {{Elsevier}},
    VOLUME = {86},
    PAGES = {21},
    YEAR = {2018},
    DOI = {10.1016/j.jsc.2017.04.001},
    KEYWORDS = {Chebyshev forms ; Chebyshev curves ; Zero dimensional systems ; Polynomial knots ; Lissajous knots ; Minimal polynomial ; Factorization of Chebyshev polynomials},
    PDF = {https://hal.inria.fr/hal-01232181/file/kprt_noels3.pdf},
    HAL_ID = {hal-01232181},
    HAL_VERSION = {v2},
    }
  • [PDF] [DOI] P. Koseleff and D. Pecker, “Harmonic knots,” Journal of knot theory and its ramifications (jktr), 2016.
    [Bibtex]
    @article{koseleff:hal-0068074,
    TITLE = {Harmonic Knots},
    AUTHOR = {Koseleff, Pierre-Vincent and Pecker, Daniel},
    URL = {https://hal.archives-ouvertes.fr/hal-00680746},
    NOTE = {18 p., 30 fig.},
    JOURNAL = {Journal Of Knot Theory And Its Ramifications (JKTR)},
    YEAR = {2016},
    DOI = {10.1142/S0218216516500747},
    KEYWORDS = {Knots ; polynomial curves ; Chebyshev curves ; rational knots ; continued fractions},
    PDF = {https://hal.archives-ouvertes.fr/hal-00680746/file/h4g5c.pdf},
    HAL_ID = {hal-00680746},
    HAL_VERSION = {v2}
    }
  • [PDF] [DOI] E. Brugallé, P. Koseleff, and D. Pecker, “Untangling trigonal diagrams,” Journal Of Knot Theory And Its Ramifications (JKTR), vol. 25, iss. 7, 2016.
    [Bibtex]
    @article{brugalle:hal-01084463,
    TITLE = {{Untangling trigonal diagrams}},
    AUTHOR = {Brugall{\'e}, Erwan and Koseleff, Pierre-Vincent and Pecker, Daniel},
    URL = {https://hal.archives-ouvertes.fr/hal-01084463},
    NOTE = {10p., 24 figs},
    JOURNAL = {{Journal Of Knot Theory And Its Ramifications (JKTR)}},
    VOLUME = {25},
    NUMBER = {7},
    YEAR = {2016},
    MONTH = Jun,
    DOI = {10.1142/S0218216516500437},
    KEYWORDS = {polynomial knots ; Two-bridge knots},
    PDF = {https://hal.archives-ouvertes.fr/hal-01084463/file/iso2.pdf},
    HAL_ID = {hal-01084463},
    HAL_VERSION = {v2}
    }
  • [PDF] [DOI] E. Brugallé, P. Koseleff, and D. Pecker, “On the lexicographic degree of two-bridge knots,” Journal Of Knot Theory And Its Ramifications (JKTR), vol. 25, iss. 7, 2016.
    [Bibtex]
    @article{brugalle:hal-01084472,
    TITLE = {{On the lexicographic degree of two-bridge knots}},
    AUTHOR = {Brugall{\'e}, Erwan and Koseleff, Pierre-Vincent and Pecker, Daniel},
    URL = {https://hal.archives-ouvertes.fr/hal-01084472},
    NOTE = {14p., 21 figs},
    JOURNAL = {{Journal Of Knot Theory And Its Ramifications (JKTR)}},
    VOLUME = {25},
    NUMBER = {7},
    YEAR = {2016},
    MONTH = Jun,
    DOI = {10.1142/S0218216516500449},
    KEYWORDS = {polynomial knots ; Real pseudoholomorphic curves ; two-bridge knots},
    PDF = {https://hal.archives-ouvertes.fr/hal-01084472/file/Lex2.pdf},
    HAL_ID = {hal-01084472},
    HAL_VERSION = {v2}
    }
  • [DOI] P. Koseleff and D. Pecker, “On alexander–conway polynomials of two-bridge links,” Journal of symbolic computation, vol. 68, Part 2, pp. 215-229, 2015.
    [Bibtex]
    @article{KP15,
    Abstract = {We consider Conway polynomials of two-bridge links as Euler continuant polynomials. As a consequence, we obtain new and elementary proofs of classical Murasugi's 1958 alternating theorem and Hartley's 1979 trapezoidal theorem. We give a modulo 2 congruence for links, which implies the classical Murasugi's 1971 congruence for knots. We also give sharp bounds for the coefficients of Euler continuants and deduce bounds for the Alexander polynomials of two-bridge links. These bounds improve and generalize those of Nakanishi--Suketa's 1996. We easily obtain some bounds for the roots of the Alexander polynomials of two-bridge links. This is a partial answer to Hoste's conjecture on the roots of Alexander polynomials of alternating knots. },
    Author = {Pierre-Vincent Koseleff and Daniel Pecker},
    Doi = {http://dx.doi.org/10.1016/j.jsc.2014.09.011},
    Issn = {0747-7171},
    Journal = {Journal of Symbolic Computation},
    Keywords = {Alexander polynomial},
    Note = {Effective Methods in Algebraic Geometry},
    Number = {0},
    Pages = {215 - 229},
    Title = {On Alexander--Conway polynomials of two-bridge links},
    Url = {http://www.sciencedirect.com/science/article/pii/S0747717114000790},
    Volume = {68, Part 2},
    Year = {2015},
    Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0747717114000790},
    Bdsk-Url-2 = {http://dx.doi.org/10.1016/j.jsc.2014.09.011}}
  • [DOI] P. V. Koseleff and D. Pecker, “Every knot is a billiard knot,” in Knots in Poland. III. Part 1, Polish Acad. Sci. Inst. Math., Warsaw, 2014, vol. 100, p. 173–178.
    [Bibtex]
    @incollection{KP14,
    Author = {Koseleff, P. V. and Pecker, D.},
    Booktitle = {Knots in {P}oland. {III}. {P}art 1},
    Doi = {10.4064/bc100-0-9},
    Mrclass = {57M25},
    Mrnumber = {3220481},
    Pages = {173--178},
    Publisher = {Polish Acad. Sci. Inst. Math., Warsaw},
    Series = {Banach Center Publ.},
    Title = {Every knot is a billiard knot},
    Url = {http://dx.doi.org/10.4064/bc100-0-9},
    Volume = {100},
    Year = {2014},
    Bdsk-Url-1 = {http://dx.doi.org/10.4064/bc100-0-9}}
  • P-V. Koseleff, D. Pecker, and F. Rouillier, “Computing chebyshev knots diagrams,” in Mega 11, 2011.
    [Bibtex]
    @Conference{KPR11b,
    Title = {Computing Chebyshev knots diagrams},
    Author = {Koseleff, P-V. and Pecker, D. and Rouillier, F.},
    Booktitle = {MEGA 11},
    Year = {2011},
    Owner = {rouillie},
    Timestamp = {2012.07.03}
    }
  • [DOI] P. -V. Koseleff and D. Pecker, “Chebyshev knots,” J. knot theory ramifications, vol. 20, iss. 4, p. 575–593, 2011.
    [Bibtex]
    @article {KP11a,
    AUTHOR = {Koseleff, P.-V. and Pecker, D.},
    TITLE = {Chebyshev knots},
    JOURNAL = {J. Knot Theory Ramifications},
    FJOURNAL = {Journal of Knot Theory and its Ramifications},
    VOLUME = {20},
    YEAR = {2011},
    NUMBER = {4},
    PAGES = {575--593},
    ISSN = {0218-2165},
    MRCLASS = {57M25 (14Q05)},
    MRNUMBER = {2796228 (2012d:57007)},
    MRREVIEWER = {Meirav Topol},
    DOI = {10.1142/S0218216511009364},
    URL = {http://dx.doi.org/10.1142/S0218216511009364},
    }
  • [DOI] P. -V. Koseleff and D. Pecker, “Chebyshev diagrams for two-bridge knots,” Geom. dedicata, vol. 150, p. 405–425, 2011.
    [Bibtex]
    @article {KP11b,
    AUTHOR = {Koseleff, P.-V. and Pecker, D.},
    TITLE = {Chebyshev diagrams for two-bridge knots},
    JOURNAL = {Geom. Dedicata},
    FJOURNAL = {Geometriae Dedicata},
    VOLUME = {150},
    YEAR = {2011},
    PAGES = {405--425},
    ISSN = {0046-5755},
    CODEN = {GEMDAT},
    MRCLASS = {14H50 (11A55 57M25)},
    MRNUMBER = {2753713 (2012g:14057)},
    DOI = {10.1007/s10711-010-9514-7},
    URL = {http://dx.doi.org/10.1007/s10711-010-9514-7},
    }
  • P. -V. Koseleff and D. Pecker, “On Fibonacci knots,” Fibonacci quart., vol. 48, iss. 2, p. 137–143, 2010.
    [Bibtex]
    @article {KP10,
    AUTHOR = {Koseleff, P.-V. and Pecker, D.},
    TITLE = {On {F}ibonacci knots},
    JOURNAL = {Fibonacci Quart.},
    FJOURNAL = {The Fibonacci Quarterly. The Official Journal of the Fibonacci
    Association},
    VOLUME = {48},
    YEAR = {2010},
    NUMBER = {2},
    PAGES = {137--143},
    ISSN = {0015-0517},
    CODEN = {FIBQAU},
    MRCLASS = {57M25 (11A55 11B39)},
    MRNUMBER = {2667798 (2011d:57015)},
    }
  • [DOI] P. -V. Koseleff, D. Pecker, and F. Rouillier, “The first rational chebyshev knots,” Journal of symbolic computation, vol. 45, iss. 12, pp. 1341-1358, 2010.
    [Bibtex]
    @Article{KPR10,
    Title = {The first rational Chebyshev knots },
    Author = {P.-V. Koseleff and D. Pecker and F. Rouillier},
    Journal = {Journal of Symbolic Computation},
    Year = {2010},
    Note = {MEGA?2009 },
    Number = {12},
    Pages = {1341 - 1358},
    Volume = {45},
    Doi = {http://dx.doi.org/10.1016/j.jsc.2010.06.014},
    ISSN = {0747-7171},
    Keywords = {Polynomial curves},
    Owner = {rouillie},
    Timestamp = {2014.01.21},
    Url = {http://www.sciencedirect.com/science/article/pii/S0747717110000945}
    }
  • [DOI] P. -V. Koseleff and D. Pecker, “A polynomial parametrization of torus knots,” Appl. algebra engrg. comm. comput., vol. 20, iss. 5-6, p. 361–377, 2009.
    [Bibtex]
    @article {KP09,
    AUTHOR = {Koseleff, P.-V. and Pecker, D.},
    TITLE = {A polynomial parametrization of torus knots},
    JOURNAL = {Appl. Algebra Engrg. Comm. Comput.},
    FJOURNAL = {Applicable Algebra in Engineering, Communication and
    Computing},
    VOLUME = {20},
    YEAR = {2009},
    NUMBER = {5-6},
    PAGES = {361--377},
    ISSN = {0938-1279},
    CODEN = {AAECEW},
    MRCLASS = {41A21 (57M25)},
    MRNUMBER = {2651586 (2011e:41027)},
    DOI = {10.1007/s00200-009-0103-7},
    URL = {http://dx.doi.org/10.1007/s00200-009-0103-7},
    }
  • [DOI] P. -V. Koseleff and D. Pecker, “On polynomial torus knots,” J. knot theory ramifications, vol. 17, iss. 12, p. 1525–1537, 2008.
    [Bibtex]
    @article {KP08,
    AUTHOR = {Koseleff, P.-V. and Pecker, D.},
    TITLE = {On polynomial torus knots},
    JOURNAL = {J. Knot Theory Ramifications},
    FJOURNAL = {Journal of Knot Theory and its Ramifications},
    VOLUME = {17},
    YEAR = {2008},
    NUMBER = {12},
    PAGES = {1525--1537},
    ISSN = {0218-2165},
    MRCLASS = {57M25 (14H50)},
    MRNUMBER = {2477592 (2009i:57014)},
    MRREVIEWER = {Swatee Naik},
    DOI = {10.1142/S0218216508006713},
    URL = {http://dx.doi.org/10.1142/S0218216508006713},
    }

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