L9a18

L9a17

L9a19

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a18 at Knotilus!
	L9a18 is is $9_{10}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a18's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3(t(1)-1)(t(2)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}}}$ (db)
Jones polynomial	$-3q^{9/2}+4q^{7/2}-4q^{5/2}+3q^{3/2}-{\frac {1}{q^{3/2}}}+q^{15/2}-2q^{13/2}+2q^{11/2}-3{\sqrt {q}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}-z^{3}a^{-5}-za^{-5}-z^{3}a^{-3}-z^{3}a^{-1}+az-za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-4}-z^{8}a^{-6}-z^{7}a^{-3}-3z^{7}a^{-5}-2z^{7}a^{-7}-z^{6}a^{-2}+3z^{6}a^{-4}+3z^{6}a^{-6}-z^{6}a^{-8}-z^{5}a^{-1}+z^{5}a^{-3}+11z^{5}a^{-5}+9z^{5}a^{-7}-4z^{4}a^{-4}-z^{4}a^{-6}+4z^{4}a^{-8}-z^{4}-az^{3}-z^{3}a^{-1}-10z^{3}a^{-5}-10z^{3}a^{-7}+3z^{2}a^{-4}-3z^{2}a^{-8}+2az+2za^{-1}+2za^{-5}+2za^{-7}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 16                   1 -1 14                 1   1 12               1 1   0 10             2 1     1 8           2 1       -1 6         2 2         0 4       1 2           1 2     2 2             0 0   1 3               2 -2                     0 -4 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L9a17 L9a19