View source for Alternating knots and links

Knots and links with an alternating diagram (cf. [[Knot and link diagrams|Knot and link diagrams]]), i.e. a projection in general position onto a plane such that, when successively traversing all components, the overpasses and underpasses succeed each other in alternation. Any diagram can be converted to an alternating diagram by changing the upper and lower branches through its double points.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a0120401.png" /> be a Seifert surface. As distinct from the ordinary case, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a0120402.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a0120403.png" /> is the degree of the Alexander polynomial (cf. [[Alexander invariants|Alexander invariants]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a0120404.png" /> is the genus of the Seifert surface and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a0120405.png" /> is the number of components of the link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a0120406.png" />, becomes an equality in the case of alternating knots and links. Accordingly, the genus of an alternating link can be computed from any one of its alternating diagrams, the Seifert surface being the surface of smallest genus. This also shows that if the diagram is normalized, i.e. if the projection plane does not contain a simple closed contour that intersects the diagram at one double point, the link is trivial (cf. [[Knot theory|Knot theory]]) if and only if the diagram contains no double points. If such a contour exists, it is possible, by rotating an internal part of the diagram through 180 degrees, to reduce the number of double points while preserving the alternating nature of the diagram. This yields an algorithm for solving the problem of triviality of alternating knots and links. Moreover, if the diagram is connected, the links do not become separated, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a0120407.png" />, and the reduced Alexander polynomial of seperated links is zero. The Alexander matrix is computed as the incidence matrix of some graph, which implies [[#References|[1]]], [[#References|[2]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a0120408.png" /> is an alternating polynomial, i.e. its coefficients are non-zero and their signs alternate. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a0120409.png" />, alternating knots and links are known as Neuwirth knots and links (cf. [[Neuwirth knot|Neuwirth knot]]). The number of double points of a normalized diagram of an alternating knot or link is not larger than its determinant. The groups of alternating knots and links (cf. [[Knot and link groups|Knot and link groups]]) are represented as free products with amalgamation of two free groups of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a01204010.png" /> by subgroups of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a01204011.png" />. This representation is obtained with the aid of the van Kampen theorem, if the space of the link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a01204012.png" /> is subdivided by the boundaries of regular neighbourhoods of the Seifert surface with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a01204013.png" />, constructed over the alternating diagram. All knots of the standard table (cf. [[Knot table|Knot table]]) with non-alternating diagrams are non-alternating knots. Most parallel knots, windings, etc., do not alternate.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Murasugi,   "On the Alexander polynomial of the alternating knot"  ''Osaka J. Math.'' , '''10'''  (1958)  pp. 181–189</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.H. Crowell,   "Genus of alternating link types"  ''Ann. of Math. (2)'' , '''69'''  (1959)  pp. 258–275</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Murasugi,   "On alternating knots"  ''Osaka J. Math.'' , '''12'''  (1960)  pp. 277–303</TD></TR></table>



====Comments====
A Seifert surface for a knot or link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a01204014.png" /> is a connected, bicollared compact manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a01204015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a01204016.png" />. A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a01204017.png" /> is bicollared (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a01204018.png" />) if there exists an imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a01204019.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a01204020.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a01204021.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012040/a01204022.png" /> or its image is the bicollar itself.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Rolfsen,   "Knots and links" , Publish or Perish  (1976)</TD></TR></table>