L10n18

L10n17

L10n19

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n18 at Knotilus!

Link Presentations

[edit Notes on L10n18's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X5,12,6,13 X8493 X13,17,14,16 X9,15,10,14 X15,11,16,10 X11,20,12,5 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {-4, -1, 2, -5, -7, 8, -9, 4, -6, 7, -8, 6, 10, -2, 3, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+2q^{5/2}-2q^{3/2}+{\sqrt {q}}-{\frac {2}{\sqrt {q}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{5}-z^{5}a^{-1}-a^{3}z^{3}+6az^{3}-5z^{3}a^{-1}+z^{3}a^{-3}-3a^{3}z+9az-8za^{-1}+2za^{-3}-a^{3}z^{-1}+4az^{-1}-4a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-2}-z^{8}-a^{3}z^{7}-2az^{7}-3z^{7}a^{-1}-2z^{7}a^{-3}-a^{4}z^{6}-a^{2}z^{6}+3z^{6}a^{-2}-z^{6}a^{-4}+4z^{6}+6a^{3}z^{5}+14az^{5}+17z^{5}a^{-1}+9z^{5}a^{-3}+5a^{4}z^{4}+8a^{2}z^{4}+4z^{4}a^{-2}+4z^{4}a^{-4}+3z^{4}-9a^{3}z^{3}-26az^{3}-26z^{3}a^{-1}-9z^{3}a^{-3}-5a^{4}z^{2}-12a^{2}z^{2}-10z^{2}a^{-2}-3z^{2}a^{-4}-14z^{2}+6a^{3}z+17az+15za^{-1}+4za^{-3}+a^{4}+4a^{2}+4a^{-2}+a^{-4}+7-a^{3}z^{-1}-4az^{-1}-4a^{-1}z^{-1}-a^{-3}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                     1 -1 8                   1   1 6                 1 1   0 4             1 2 1     0 2           1 1 1       1 0         1 4 2         1 -2       1 1 2           2 -4       1 1             0 -6   1 1 1               1 -8                       0 -10 1                     -1

Integral Khovanov Homology (db, data source)

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

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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