L11n5

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n5 at Knotilus!

Link Presentations

[edit Notes on L11n5's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X17,1,18,4 X9,14,10,15 X3849 X5,11,6,10 X11,19,12,18 X13,20,14,21 X19,5,20,22 X21,12,22,13 X2,16,3,15
Gauss code	{1, -11, -5, 3}, {-6, -1, 2, 5, -4, 6, -7, 10, -8, 4, 11, -2, -3, 7, -9, 8, -10, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)^{3}}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{7/2}-3q^{5/2}+3q^{3/2}-5{\sqrt {q}}+{\frac {5}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-2a^{3}z^{3}-5a^{3}z-3a^{3}z^{-1}+za^{-3}+az^{5}+4az^{3}-2z^{3}a^{-1}+7az+4az^{-1}-4za^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-5a^{2}z^{8}-3z^{8}-a^{5}z^{7}+a^{3}z^{7}-az^{7}-3z^{7}a^{-1}+10a^{4}z^{6}+24a^{2}z^{6}-z^{6}a^{-2}+13z^{6}+5a^{5}z^{5}+13a^{3}z^{5}+21az^{5}+13z^{5}a^{-1}-14a^{4}z^{4}-33a^{2}z^{4}+z^{4}a^{-2}-18z^{4}-7a^{5}z^{3}-26a^{3}z^{3}-37az^{3}-21z^{3}a^{-1}-3z^{3}a^{-3}+7a^{4}z^{2}+17a^{2}z^{2}-z^{2}a^{-4}+11z^{2}+4a^{5}z+16a^{3}z+22az+13za^{-1}+3za^{-3}-a^{4}-3a^{2}-a^{-2}-2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 8                   1 -1 6                 2   2 4               2 2   0 2             4 2     2 0           3 4 1     0 -2         3 4 1       0 -4       2 3           1 -6     1 3 1           -1 -8   1 2               1 -10   1                 -1 -12 1                   1

Integral Khovanov Homology (db, data source)

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

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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