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。

亚历山大–康威多项式

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

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

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

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. 
• 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). 
• 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