亚历山大多项式

在纽结理论中，亚历山大多项式（Alexander polynomial）是一种纽结多项式。^[1]

亚历山大–康威多项式

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

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

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

参考文献

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.

阅读

• Alexander, J. W. Topological invariants of knots and links. Transactions of the American Mathematical Society. 1928, 30 (2): 275–306. JSTOR 1989123. doi:10.2307/1989123.
• Crowell, Richard; Fox, Ralph. Introduction to Knot Theory. Ginn and Co. after 1977 Springer Verlag. 1963.
• Adams, Colin C. The Knot Book: An elementary introduction to the mathematical theory of knots 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. A quick trip through knot theory, In Topology of ThreeManifold 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. Topology of 4-manifolds. Princeton Mathematical Series 39. Princeton, NJ: Princeton University Press. 1990. ISBN 978-0-691-08577-7.
• Kauffman, Louis. Formal Knot Theory. Princeton University Press. 1983.
• Kauffman, Louis. Knots and Physics 4th. World Scientific Publishing Company. 2012. ISBN 978-981-4383-00-4.
• Kawauchi, Akio. A Survey of Knot Theory. Birkhauser. 1996. (covers several different approaches, explains relations between different versions of the Alexander polynomial)
• Ozsváth, Peter; Szabó, Zoltán. Holomorphic disks and knot invariants. Advances in Mathematics. 2004, 186 (1): 58–116. Bibcode:2002math......9056O. arXiv:math/0209056 . doi:10.1016/j.aim.2003.05.001.
• Szabó, Zoltán. Holomorphic disks and genus bounds. 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. Knot Floer homology detects fibred knots. 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. (学位论文). 缺少或|title=为空 (帮助)
• Rolfsen, Dale. Knots and Links 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), Alexander invariants, 数学百科全书, 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