L9n8

L9n7

L9n9

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n8 at Knotilus!
	L9n8 is $9_{56}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n8's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X8493 X9,14,10,15 X13,10,14,11 X11,18,12,5 X2,16,3,15
Gauss code	{1, -9, 5, -3}, {-4, -1, 2, -5, -6, 7, -8, 4, -7, 6, 9, -2, 3, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	${\frac {4}{q^{9/2}}}-{\frac {5}{q^{7/2}}}+{\frac {3}{q^{5/2}}}-{\frac {4}{q^{3/2}}}+{\frac {2}{q^{13/2}}}-{\frac {3}{q^{11/2}}}-{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{7}z^{-1}-z^{3}a^{5}+2a^{5}z^{-1}+z^{5}a^{3}+3z^{3}a^{3}+2za^{3}-z^{3}a-2za-az^{-1}$ (db)
Kauffman polynomial	$-3z^{2}a^{8}+2a^{8}-z^{5}a^{7}-z^{3}a^{7}-a^{7}z^{-1}-2z^{6}a^{6}+4z^{4}a^{6}-8z^{2}a^{6}+5a^{6}-z^{7}a^{5}-z^{5}a^{5}+4z^{3}a^{5}-za^{5}-2a^{5}z^{-1}-4z^{6}a^{4}+9z^{4}a^{4}-6z^{2}a^{4}+3a^{4}-z^{7}a^{3}-z^{5}a^{3}+8z^{3}a^{3}-4za^{3}-2z^{6}a^{2}+5z^{4}a^{2}-z^{2}a^{2}-a^{2}-z^{5}a+3z^{3}a-3za+az^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 χ 2               1 1 0             1   -1 -2           3 1   2 -4         2 3     1 -6       3 1       2 -8     1 2         1 -10   2 3           -1 -12   1             1 -14 2               -2

Integral Khovanov Homology (db, data source)

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

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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