Knot Genus

Website Links For Seifert

Information About Knot Genus

	CATEGORIES ABOUT SEIFERT SURFACE
	geometric topology
	knot theory
	surfaces

	APPAREL
	BABY
	BEAUTY
	BOOKS
	CAR TOYS
	CELL PHONES
	DVD'S
	ELECTRONICS
	GOURMET FOOD
	GROCERIES
	HEALTH & PERSONAL
	HOME & GARDEN
	JEWELRY
	MUSIC
	MUSIC INSTRUMENTS
	OFFICE PRODUCTS
	SOFTWARE
	SPORTING GOODS
	TOOLS & HARDWARE
	TOYS
	VIDEO GAMES
	SHOPPING HOME
	MORE SHOPPING...

Specifically, let ''L'' be a Tame Oriented knot or link in Euclidean 3-space (or in the 3-sphere ). A Seifert surface is a Compact , Connected , Oriented Surface ''S'' embedded in 3-space whose boundary is ''L'' such that the orientation on ''L'' is just the induced orientation from ''S''. It is a Theorem that there always exists such a surface. This theorem was first published by F. Frankl and Lev Pontrjagin in 1930 . A different proof was published in 1934 by Herbert Seifert and relies on what is now called the Seifert algorithm. The Algorithm produces a Seifert surface, given a knot or link diagram. The algorithm also computes the 2g$imes$2g integer Seifert Matrix A of the linking numbers of the 2g linearly independent cycles generating the torsion-free quotient of the homology group of the Seifert surface, where g is the Genus of the surface; the (i,j) entry in A is the linking number in Euclidean 3-space (or in the 3-sphere ) of the ith cycle and the pushoff of the jth cycle out of the surface. Note that any compact, connected, oriented surface with nonempty boundary is the Seifert surface associated to its boundary link. A single knot or link can have many different inequivalent Seifert surfaces. It is important to note that a Seifert surface must be ''oriented''. It is possible to associated unoriented surfaces to knots as well. EXAMPLES The standard Möbius Strip has the Unknot for a boundary but is not considered to be a Seifert surface for the unknot because it is not orientable. The "checkerboard" coloring of the minimal crossing projection of the Trefoil Knot gives a Mobius strip with three half twists. As with the previous example, this is not a Seifert surface as it is not orientable. Applying Seifert's algorithm to this diagram, as expected, does produce a Seifert surface; in this case, it is a punctured torus of genus g=1, and the Seifert matrix is

The SeifertView programme http://www.win.tue.nl/~vanwijk/seifertview/ of Jack van Wijk visualizes
the Seifert surfaces of knots constructed using Seifert's algorithm.


GENUS OF A KNOT


One example of a Knot Invariant which is computed from a Seifert surface is the genus of a knot. The genus of a knot ''K'' is defined as minimal Genus g of all Seifert surfaces for ''K''.

For instance:

• An Unknot — which is, by definition, the boundary of a Disc — has genus zero. Moreover, the unknot is the ''only'' knot with genus zero.


• The Trefoil Knot has genus one, as does the Figure-eight Knot .


• The genus of a (''p'',''q'')- Torus Knot is (''p'' − 1)(''q'' − 1)/2

A fundamental property of the genus is that it is additive with respect to the Knot Sum :
:$g(K\_1\; \backslash \#\; K\_2)\; =\; g(K\_1)\; +\; g(K\_2)$


CROSSCAP NUMBER OF A KNOT


Every knot ''K'' in 3-space bounds a compact, connected non-orientable surface of genus
$g$. The minimum of such $g$ is called the Crosscap Number or
non-orientable genus of ''K''.

For instance:

• The Crosscap number of the Unknot is defined to be Zero.


• The Crosscap number of the trefoil is One, as it bounds a Möbius Strip and is not trivial.


• The Crosscap number of a Torus Knot was determined by M.Teragaito.

The formula for the Knot Sum is
:$C(k\_1)+C(k\_2)-1\; \backslash leq\; C(k\_1\; \backslash \#\; k\_2)\; \backslash leq\; C(k\_1)+C(k\_2).$


REFERENCES

• Clark, B.E. "Crosscaps and Knots" , Int. J. Math and Math. Sci, Vol 1, 1978, pp 113-124


• Murakami, Hitoshi and Yasuhara, Akira. "Crosscap number of a knot," Pacific J. Math. 171 (1995), no. 1, 261--273.


• Teragaito, Masakazu. "Crosscap numbers of torus knots," Topology Appl. 138 (2004), no. 1-3, 219--238.


• Teragaito, Masakazu and Hirasawa, Mikami. "Crosscap numbers of 2-bridge knots," Arxiv:math.GT/0504446.


• J.Uhing. "Zur Kreuzhaubenzahl von Knoten" , diploma thesis, 1997, University of Dortmund, (in german)