L11n9

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n9 at Knotilus!

Link Presentations

[edit Notes on L11n9's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X8493 X5,11,6,10 X11,20,12,21 X19,22,20,5 X13,19,14,18 X21,12,22,13 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 6, -7, 10, -9, 4, 11, -2, 3, 9, -8, 7, -10, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {4}{q^{7/2}}}-q^{5/2}+{\frac {3}{q^{5/2}}}+2q^{3/2}-{\frac {4}{q^{3/2}}}+{\frac {1}{q^{13/2}}}-{\frac {1}{q^{11/2}}}-3{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{7}z^{-1}+2a^{5}z+3a^{5}z^{-1}-2a^{3}z^{3}-5a^{3}z-3a^{3}z^{-1}+az^{5}+4az^{3}-z^{3}a^{-1}+5az+2az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{2}-a^{8}+a^{7}z^{3}-a^{7}z+a^{7}z^{-1}+a^{6}z^{4}-2a^{6}+a^{5}z^{7}-4a^{5}z^{5}+6a^{5}z^{3}-6a^{5}z+3a^{5}z^{-1}+2a^{4}z^{8}-10a^{4}z^{6}+15a^{4}z^{4}-8a^{4}z^{2}+a^{3}z^{9}-2a^{3}z^{7}-8a^{3}z^{5}+18a^{3}z^{3}-12a^{3}z+3a^{3}z^{-1}+4a^{2}z^{8}-20a^{2}z^{6}+27a^{2}z^{4}-12a^{2}z^{2}+2a^{2}+az^{9}-2az^{7}+z^{7}a^{-1}-9az^{5}-5z^{5}a^{-1}+20az^{3}+7z^{3}a^{-1}-11az-4za^{-1}+2az^{-1}+a^{-1}z^{-1}+2z^{8}-10z^{6}+13z^{4}-5z^{2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                   1 1 4                 1   -1 2               2 1   1 0           1 2 1     0 -2         1 4 2       1 -4       1 3 3         1 -6       3 2           1 -8   1 2 3             2 -10   1 2               -1 -12 1 1                 0 -14 1                   -1

Integral Khovanov Homology (db, data source)

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

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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