L9a11

L9a10

L9a12

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a11 at Knotilus!
	L9a11 is $9_{26}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a11's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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