L9a1

L8n8

L9a2

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a1 at Knotilus!
	L9a1 is $9_{32}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a1's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X10,6,11,5 X8493 X18,12,5,11 X12,18,13,17 X16,10,17,9 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 $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{13/2}+3q^{11/2}-5q^{9/2}+8q^{7/2}-10q^{5/2}+9q^{3/2}-9{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}+z^{5}a^{-3}-az^{3}+z^{3}a^{-1}+2z^{3}a^{-3}-z^{3}a^{-5}-za^{-1}+2za^{-3}-za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{8}a^{-2}-2z^{8}a^{-4}-5z^{7}a^{-1}-9z^{7}a^{-3}-4z^{7}a^{-5}-5z^{6}a^{-2}-2z^{6}a^{-4}-3z^{6}a^{-6}-6z^{6}-4az^{5}+4z^{5}a^{-1}+18z^{5}a^{-3}+9z^{5}a^{-5}-z^{5}a^{-7}-a^{2}z^{4}+11z^{4}a^{-2}+10z^{4}a^{-4}+7z^{4}a^{-6}+7z^{4}+4az^{3}-13z^{3}a^{-3}-7z^{3}a^{-5}+2z^{3}a^{-7}-4z^{2}a^{-2}-7z^{2}a^{-4}-4z^{2}a^{-6}-z^{2}+2za^{-1}+4za^{-3}+2za^{-5}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 14                   1 1 12                 2   -2 10               3 1   2 8             5 2     -3 6           5 3       2 4         4 5         1 2       5 5           0 0     3 6             3 -2   1 3               -2 -4   3                 3 -6 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

L8n8 L9a2