L9a26

L9a25

L9a27

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a26 at Knotilus!
	L9a26 is $9_{11}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a26's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X18,10,7,9 X14,6,15,5 X16,14,17,13 X12,18,13,17 X2738 X4,12,5,11 X6,16,1,15
Gauss code	{1, -7, 2, -8, 4, -9}, {7, -1, 3, -2, 8, -6, 5, -4, 9, -5, 6, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{2}v^{2}-3u^{2}v+u^{2}-3uv^{2}+5uv-3u+v^{2}-3v+2}{uv}}$ (db)
Jones polynomial	$8q^{9/2}-8q^{7/2}+6q^{5/2}-5q^{3/2}+q^{17/2}-3q^{15/2}+5q^{13/2}-7q^{11/2}+2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+za^{-7}-z^{5}a^{-5}-2z^{3}a^{-5}-za^{-5}-z^{5}a^{-3}-2z^{3}a^{-3}-za^{-3}-a^{-3}z^{-1}+z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}-z^{2}a^{-10}+3z^{5}a^{-9}-4z^{3}a^{-9}+za^{-9}+4z^{6}a^{-8}-5z^{4}a^{-8}+2z^{2}a^{-8}+3z^{7}a^{-7}-z^{5}a^{-7}-2z^{3}a^{-7}+za^{-7}+z^{8}a^{-6}+5z^{6}a^{-6}-9z^{4}a^{-6}+4z^{2}a^{-6}+5z^{7}a^{-5}-6z^{5}a^{-5}+za^{-5}+z^{8}a^{-4}+3z^{6}a^{-4}-7z^{4}a^{-4}+2z^{2}a^{-4}+2z^{7}a^{-3}-z^{5}a^{-3}-5z^{3}a^{-3}+4za^{-3}-a^{-3}z^{-1}+2z^{6}a^{-2}-4z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}+z^{5}a^{-1}-3z^{3}a^{-1}+3za^{-1}-a^{-1}z^{-1}$ (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 18                   1 -1 16                 2   2 14               3 1   -2 12             4 2     2 10           4 3       -1 8         4 4         0 6       3 5           2 4     2 3             -1 2   1 4               3 0   1                 -1 -2 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L9a25 L9a27