L9a8

L9a7

L9a9

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a8 at Knotilus!
	L9a8 is $9_{25}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a8's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X16,8,17,7 X18,13,5,14 X14,17,15,18 X12,10,13,9 X8,16,9,15 X2536 X4,12,1,11
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -7, 6, -2, 9, -6, 4, -5, 7, -3, 5, -4}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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