L11n129

L11n128

L11n130

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n129 at Knotilus!

Link Presentations

[edit Notes on L11n129's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X4,15,5,16 X5,13,6,12 X16,12,17,11 X17,6,18,1 X19,14,20,15 X13,20,14,21 X21,19,22,18
Gauss code	{1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, 7, 6, -10, 9, 5, -7, -8, 11, -9, 10, -11, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{2}-2u^{2}v+u^{2}-3uv^{2}+7uv-3u+v^{2}-2v+1}{uv}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {1}{q^{9/2}}}-3q^{7/2}+{\frac {2}{q^{7/2}}}+4q^{5/2}-{\frac {5}{q^{5/2}}}-6q^{3/2}+{\frac {6}{q^{3/2}}}+7{\sqrt {q}}-{\frac {7}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}+z^{3}a^{-3}-3a^{3}z-2a^{3}z^{-1}+za^{-3}-z^{5}a^{-1}+3az^{3}-3z^{3}a^{-1}+5az+2az^{-1}-4za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-2a^{2}z^{8}-3z^{8}a^{-2}-5z^{8}-a^{3}z^{7}-2z^{7}a^{-1}-3z^{7}a^{-3}+8a^{2}z^{6}+10z^{6}a^{-2}-z^{6}a^{-4}+19z^{6}+3a^{3}z^{5}+10az^{5}+18z^{5}a^{-1}+11z^{5}a^{-3}-2a^{4}z^{4}-15a^{2}z^{4}-7z^{4}a^{-2}+3z^{4}a^{-4}-23z^{4}-a^{5}z^{3}-9a^{3}z^{3}-21az^{3}-22z^{3}a^{-1}-9z^{3}a^{-3}+2a^{4}z^{2}+8a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+9z^{2}+2a^{5}z+8a^{3}z+12az+8za^{-1}+2za^{-3}-a^{2}-a^{5}z^{-1}-2a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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 10                   1 -1 8                 2   2 6               2 1   -1 4             4 2     2 2           3 2       -1 0         4 4         0 -2       3 4           1 -4     2 3             -1 -6   1 4               3 -8   1                 -1 -10 1                   1

Integral Khovanov Homology (db, data source)

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

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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