Alternating Knot

An alternating knot is a Knot which possesses a knot diagram in which crossings alternate between under- and overpasses. Not all knot diagrams of alternating knots need be alternating diagrams.

The Trefoil Knot and Figure-of-Eight Knot are alternating knots. One of Tait's Knot Conjectures states that the number of crossings is the same for any diagram of a reduced alternating knot. Furthermore, a reduced alternating projection of a knot has the least number of crossings for any projection of that knot. Both of these facts were proved true by Kauffman (1988), Thistlethwaite (1987), and Murasugi (1987).

If $K$ has a reduced alternating projection of $n$ crossings, then the Span of $K$ is $4n$. Let $c(K)$ be the Crossing Number. Then an alternating knot $K_1\char93 K_2$ (a Knot Sum) satisfies

$$c(K_1\char93 K_2)=c(K_1)+c(K_2).$$

In fact, this is true as well for the larger class of Adequate Knots and postulated for all Knots. The number of Prime alternating knots of $n$ crossing for $n=1$, 2, ... are 0, 0, 1, 1, 2, 3, 7, 18, 41, 123, 367, ... (Sloane's A002864).

See also Adequate Knot, Almost Alternating Link, Alternating Link, Flyping Conjecture

References

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 159-164, 1994.

Arnold, B.; Au, M.; Candy, C.; Erdener, K.; Fan, J.; Flynn, R.; Muir, J.; Wu, D.; and Hoste, J. ``Tabulating Alternating Knots through 14 Crossings.'' ftp://chs.cusd.claremont.edu/pub/knot/paper.TeX.txt.

Arnold, B.; Au, M.; Candy, C.; Erdener, K.; Fan, J.; Flynn, R.; Muir, J.; Wu, D.; and Hoste, J. ftp://chs.cusd.claremont.edu/pub/knot/AltKnots/.

Erdener, K. and Flynn, R. ``Rolfsen's Table of all Alternating Diagrams through 9 Crossings.'' ftp://chs.cusd.claremont.edu/pub/knot/Rolfsen_table.final.

Kauffman, L. ``New Invariants in the Theory of Knots.'' Amer. Math. Monthly 95, 195-242, 1988.

Murasugi, K. ``Jones Polynomials and Classical Conjectures in Knot Theory.'' Topology 26, 297-307, 1987.

Sloane, N. J. A. Sequence A002864/M0847 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Thistlethwaite, M. ``A Spanning Tree Expansion for the Jones Polynomial.'' Topology 26, 297-309, 1987.