L9n2

L9n1

L9n3

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n2 at Knotilus!
	L9n2 is $9_{46}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n2's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

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

Integral Khovanov Homology (db, data source)

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

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.

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