亞歷山大多項式

在纽结理论中，亚历山大多项式（Alexander polynomial）是一种紐結多項式。[1]

亚历山大–康威多项式

$\nabla (O)=1$ (unknot)
$\nabla (L_{+})-\nabla (L_{-})=z\nabla (L_{0})$

$\Delta _{L}(t^{2})=\nabla _{L}(t-t^{-1})$

$\Delta (L_{+})-\Delta (L_{-})=(t^{1/2}-t^{-1/2})\Delta (L_{0})$

参考文献

Alexander describes his skein relation toward the end of his paper under the heading "miscellaneous theorems", which is possibly why it got lost. Joan Birman mentions in her paper New points of view in knot theory (Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 253–287) that Mark Kidwell brought her attention to Alexander's relation in 1970.

阅读

Alexander, J. W. . Transactions of the American Mathematical Society. 1928, 30 (2): 275–306. JSTOR 1989123. doi:10.2307/1989123.
Crowell, Richard; Fox, Ralph. . Ginn and Co. after 1977 Springer Verlag. 1963.
Adams, Colin C. Revised reprint of the 1994 original. Providence, RI: American Mathematical Society. 2004. ISBN 978-0-8218-3678-1. (accessible introduction utilizing a skein relation approach)
Fox, Ralph. Proceedings of 1961 Topology Institute at Univ. of Georgia, edited by M.K.Fort. Englewood Cliffs. N. J.: Prentice-Hall: 120–167. 1961.
Freedman, Michael H.; Quinn, Frank. . Princeton Mathematical Series 39. Princeton, NJ: Princeton University Press. 1990. ISBN 978-0-691-08577-7.
Kauffman, Louis. . Princeton University Press. 1983.
Kauffman, Louis. 4th. World Scientific Publishing Company. 2012. ISBN 978-981-4383-00-4.
Kawauchi, Akio. . Birkhauser. 1996. (covers several different approaches, explains relations between different versions of the Alexander polynomial)
Ozsváth, Peter; Szabó, Zoltán. . Advances in Mathematics. 2004, 186 (1): 58–116. Bibcode:2002math......9056O. arXiv:math/0209056. doi:10.1016/j.aim.2003.05.001.
Szabó, Zoltán. . Geometry and Topology. 2004b, 8 (2004): 311–334. arXiv:math/0311496. doi:10.2140/gt.2004.8.311. Authors list列表中的|first1=缺少|last1= (帮助)
Ni, Yi. . Inventiones Mathematicae. Invent. Math. 2007, 170 (3): 577–608. Bibcode:2007InMat.170..577N. arXiv:math/0607156. doi:10.1007/s00222-007-0075-9.
Rasmussen, Jacob. (Thesis). 缺少或|title=为空 (帮助)
Rolfsen, Dale. 2nd. Berkeley, CA: Publish or Perish. 1990. ISBN 978-0-914098-16-4. (explains classical approach using the Alexander invariant; knot and link table with Alexander polynomials)

外部链接

Hazewinkel, Michiel (编), , , Springer, 2001, ISBN 978-1-55608-010-4
"Main Page" and "The Alexander-Conway Polynomial", The Knot Atlas. – knot and link tables with computed Alexander and Conway polynomials

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[1] Alexander describes his skein relation toward the end of his paper under the heading "miscellaneous theorems", which is possibly why it got lost. Joan Birman mentions in her paper New points of view in knot theory (Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 253–287) that Mark Kidwell brought her attention to Alexander's relation in 1970.