L9n17

L9n16

L9n18

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n17 at Knotilus!
	L9n17 is $9_{52}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n17's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X7,14,8,15 X13,18,14,7 X17,1,18,6 X16,11,17,12 X5,12,6,13 X4,16,5,15
Gauss code	{1, -2, 3, -9, -8, 6}, {-4, -1, 2, -3, 7, 8, -5, 4, 9, -7, -6, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} , ...)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{t(2)^2 t(1)^2-t(2) t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-2 t(1)-t(2)+1}{t(1) t(2)}} (db)
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{4}{q^{9/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{2}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{2}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{4}{q^{11/2}}} (db)
Signature	-3 (db)
HOMFLY-PT polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^3 a^7-2 z a^7-a^7 z^{-1} +z^5 a^5+4 z^3 a^5+6 z a^5+3 a^5 z^{-1} -2 z^3 a^3-5 z a^3-2 a^3 z^{-1} } (db)
Kauffman polynomial	$-z^{4}a^{10}+2z^{2}a^{10}-2z^{5}a^{9}+4z^{3}a^{9}-za^{9}-2z^{6}a^{8}+4z^{4}a^{8}-3z^{2}a^{8}+a^{8}-z^{7}a^{7}+z^{5}a^{7}-2z^{3}a^{7}+2za^{7}-a^{7}z^{-1}-3z^{6}a^{6}+7z^{4}a^{6}-9z^{2}a^{6}+3a^{6}-z^{7}a^{5}+3z^{5}a^{5}-9z^{3}a^{5}+9za^{5}-3a^{5}z^{-1}-z^{6}a^{4}+2z^{4}a^{4}-4z^{2}a^{4}+3a^{4}-3z^{3}a^{3}+6za^{3}-2a^{3}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed $j$, alternation over $r$).

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 χ -2               2 2 -4             1 1 0 -6           3 1   2 -8         2 2     0 -10       2 2       0 -12     1 2         1 -14   1 2           -1 -16   1             1 -18 1               -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

L9n16 L9n18