L9n3

L9n2

L9n4

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n3 at Knotilus!
	L9n3 is $9_{47}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n3's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 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 $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}-{\frac {2}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {1}{q^{9/2}}}+{\frac {1}{q^{11/2}}}-{\frac {1}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+a^{5}z^{3}+3a^{5}z+2a^{5}z^{-1}-a^{3}z-az-az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-5a^{8}z^{4}+6a^{8}z^{2}-2a^{8}+a^{7}z^{7}-5a^{7}z^{5}+6a^{7}z^{3}-2a^{7}z+a^{7}z^{-1}+2a^{6}z^{6}-10a^{6}z^{4}+13a^{6}z^{2}-5a^{6}+a^{5}z^{7}-5a^{5}z^{5}+7a^{5}z^{3}-5a^{5}z+2a^{5}z^{-1}+a^{4}z^{6}-5a^{4}z^{4}+7a^{4}z^{2}-3a^{4}+a^{3}z^{3}-2a^{3}z+a^{2}+az-az^{-1}$ (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               1 1 -2             1 2 1 -4           1   1 2 -6         1 2     1 -8       1 1       0 -10       1 1       0 -12   1 1           0 -14                 0 -16 1               -1

Integral Khovanov Homology (db, data source)

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

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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