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A PARTIAL ORDER ON THE SET OF PRIME KNOTS WITH UP TO 11 CROSSINGS

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- ERRATA: "A PARTIAL ORDER ON THE SET OF PRIME KNOTS WITH UP TO 11 CROSSINGS"

DeNA Co., Ltd, Japan

Department of Information Systems Science, Faculty of Engineering, Soka University, 1-236 Tangi-cho, Hachioji-shi, Tokyo, 192-8577, Japan

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MINEKO MATSUMOTO

Division of Information Systems Science, Graduate School of Engineering, Soka University, 1-236 Tangi-cho, Hachioji-shi, Tokyo, 192-8577, Japan

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, and

MASAAKI SUZUKI

Department of Mathematics, Akita University, 1-1 TegataGakuenmachi, Akita 010-8502, Japan

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https://doi.org/10.1142/S0218216511008747Cited by:12 (Source: Crossref)

Abstract

Let K be a prime knot in S³ and G(K) = π₁(S³ - K) the knot group. We write K₁ ≥ K₂ if there exists a surjective homomorphism from G(K₁) onto G(K₂). In this paper, we determine this partial order on the set of prime knots with up to 11 crossings. There exist such 801 prime knots and then 640, 800 should be considered. The existence of a surjective homomorphism can be proved by constructing it explicitly. On the other hand, the non-existence of a surjective homomorphism can be proved by the Alexander polynomial and the twisted Alexander polynomial.

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Keywords:

AMSC: 57M25, 57M05, 57M27

References

M. Boileauet al., Comm. Anal. Geom. 18, 121 (2010). Crossref, Web of Science, Google Scholar
R. Crowell and R. Fox , Introduction to Knot Theory , Graduate Texts in Mathematics 57 ( Springer ) . Google Scholar
P. Gilmer, Topol. Appl. 18, 313 (1984), DOI: 10.1016/0166-8641(84)90016-6. Crossref, Web of Science, Google Scholar
C. Gordon, Math. Ann. 257, 157 (1981), DOI: 10.1007/BF01458281. Crossref, Web of Science, Google Scholar
B. Jiang and S. Wang, Topics in Knot Theory, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences 399 (Kluwer Academic Publishers, Dordrecht, 1993) pp. 211–227. Crossref, Google Scholar
R. Kirby, Geometric Topology, AMS/IP Studies in Advanced Mathematics 2.2 (American Mathematical Society, Providence, 1997) pp. 35–473. Google Scholar
T. Kitano, M. Suzuki and M. Wada, Algebr. Geom. Topol. 5, 1315 (2005), DOI: 10.2140/agt.2005.5.1315. Crossref, Web of Science, Google Scholar
T. Kitano and M. Suzuki, Exp. Math. 14, 385 (2005), DOI: 10.1080/10586458.2005.10128937. Crossref, Web of Science, Google Scholar
T. Kitano and M. Suzuki, Intelligence of Low Dimensional Topology (World Scientific, 2006) pp. 173–178. Google Scholar
T. Kitano and M. Suzuki, Geometry and Topology Monographs 13 (2008) pp. 307–322. Crossref, Google Scholar
T. Kitano and M. Suzuki, Acta Math. Sinica 24, 1801 (2008), DOI: 10.1007/s10114-008-6269-2. Crossref, Web of Science, Google Scholar
K. Kodama, http:/-2pt/www.math.kobe-u.ac.jp/HOME/kodama/knot.html . Google Scholar
X. S. Lin, Acta Math. Sin. (Engl. Ser.) 17, 361 (2001), DOI: 10.1007/s101140100122. Crossref, Web of Science, Google Scholar
J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants, http:/-2pt/www.indiana.edu/, March 16, 2009 . Google Scholar
T. Ohtsuki, R. Riley and M. Sakuma, Geometry and Topology Monographs 14 (2008) pp. 417–450. Crossref, Google Scholar
D. Rolfsen , Knots and Links ( Publish or Perish, Inc ) . Google Scholar
M. Wada, Topology 33, 241 (1994), DOI: 10.1016/0040-9383(94)90013-2. Crossref, Web of Science, Google Scholar
M. Wada, J. Knot Theory Ramifications 2, 233 (1993), DOI: 10.1142/S0218216593000143. Link, Google Scholar
S. Wang, Non-zero degree maps between 3-manifolds, Proceedings of the International Congress of MathematiciansII (Higher Education Press, Beijing, 2002) pp. 457–468. Google Scholar
Wolfram Research, Mathematica . Google Scholar