L11n208

L11n207

L11n209

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n208 at Knotilus!

Link Presentations

[edit Notes on L11n208's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X5,14,6,15 X22,18,9,17 X19,5,20,4 X21,6,22,7 X7,17,8,16 X8,9,1,10 X18,14,19,13 X15,21,16,20
Gauss code	{1, -2, 3, 6, -4, 7, -8, -9}, {9, -1, 2, -3, 10, 4, -11, 8, 5, -10, -6, 11, -7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{3}v^{3}-u^{3}v^{2}+u^{2}v^{2}+uv-v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}+q^{7/2}-{\frac {1}{q^{7/2}}}-q^{5/2}+{\frac {1}{q^{5/2}}}+q^{3/2}-{\frac {2}{q^{3/2}}}-{\frac {1}{q^{11/2}}}-2{\sqrt {q}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{3}z^{5}+5a^{3}z^{3}+6a^{3}z+2a^{3}z^{-1}-az^{7}-7az^{5}+z^{5}a^{-1}-16az^{3}+5z^{3}a^{-1}-14az+6za^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-a^{2}z^{8}-z^{8}a^{-2}-2z^{8}+7az^{7}+7z^{7}a^{-1}+6a^{2}z^{6}+7z^{6}a^{-2}+13z^{6}-2a^{3}z^{5}-18az^{5}-16z^{5}a^{-1}-a^{4}z^{4}-12a^{2}z^{4}-15z^{4}a^{-2}-26z^{4}-a^{5}z^{3}+8a^{3}z^{3}+24az^{3}+15z^{3}a^{-1}-a^{6}z^{2}+2a^{4}z^{2}+11a^{2}z^{2}+10z^{2}a^{-2}+18z^{2}-a^{7}z+a^{5}z-7a^{3}z-16az-7za^{-1}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 8                   1 -1 6                     0 4               1 1   0 2             1       1 0             1       1 -2         2 1         1 -4         1           1 -6     1 1             0 -8                     0 -10 1 1                 0 -12 1                   1

Integral Khovanov Homology (db, data source)

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

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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