L9n5

L9n4

L9n6

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n5 at Knotilus!
	L9n5 is $9_{44}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n5's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X15,1,16,4 X5,12,6,13 X3849 X9,16,10,17 X11,18,12,5 X17,10,18,11 X2,14,3,13
Gauss code	{1, -9, -5, 3}, {-4, -1, 2, 5, -6, 8, -7, 4, 9, -2, -3, 6, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in 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 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} , $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {3}{q^{9/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{5/2}}}-{\frac {2}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{15/2}}}+{\frac {2}{q^{13/2}}}-{\frac {3}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-3za^{7}-2a^{7}z^{-1}+z^{5}a^{5}+5z^{3}a^{5}+9za^{5}+5a^{5}z^{-1}-2z^{3}a^{3}-6za^{3}-3a^{3}z^{-1}$ (db)
Kauffman 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 a^{10} z^4-3 a^{10} z^2+a^{10}+a^9 z^5-2 a^9 z^3+a^8 z^6-2 a^8 z^4+a^8 z^2+a^7 z^7-4 a^7 z^5+8 a^7 z^3-5 a^7 z+2 a^7 z^{-1} +2 a^6 z^6-7 a^6 z^4+12 a^6 z^2-5 a^6+a^5 z^7-5 a^5 z^5+13 a^5 z^3-13 a^5 z+5 a^5 z^{-1} +a^4 z^6-4 a^4 z^4+8 a^4 z^2-5 a^4+3 a^3 z^3-8 a^3 z+3 a^3 z^{-1} } (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ 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 2 1 -6           2     2 -8         1 1     0 -10       2 2       0 -12       1         1 -14   1 2           -1 -16                 0 -18 1               -1

Integral Khovanov Homology (db, data source)

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

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.

L9n4 L9n6