L10n109

L10n108

L10n110

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n109 at Knotilus!

Link Presentations

[edit Notes on L10n109's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(2)t(3)^{2}+t(2)t(4)t(3)^{2}-t(4)t(3)^{2}+t(1)t(4)^{2}t(3)-t(1)t(2)t(4)^{2}t(3)+t(2)t(3)-t(1)t(4)t(3)+2t(1)t(2)t(4)t(3)-t(2)t(4)t(3)+2t(4)t(3)-t(3)-t(1)t(4)^{2}+t(1)t(4)-t(1)t(2)t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$-2q^{3/2}+2{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {4}{q^{3/2}}}-{\frac {7}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z^{-3}+2a^{5}z+2a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}-3a^{3}z^{-3}-8a^{3}z-7a^{3}z^{-1}+3az^{3}+3az^{-3}-a^{-1}z^{-3}+8az+8az^{-1}-2za^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+a^{7}z+2a^{6}z^{6}-5a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-11a^{5}z^{5}+15a^{5}z^{3}-a^{5}z^{-3}-12a^{5}z+5a^{5}z^{-1}+a^{4}z^{8}+a^{4}z^{6}-11a^{4}z^{4}+17a^{4}z^{2}+3a^{4}z^{-2}-10a^{4}+5a^{3}z^{7}-20a^{3}z^{5}+36a^{3}z^{3}-3a^{3}z^{-3}-29a^{3}z+12a^{3}z^{-1}+a^{2}z^{8}-8a^{2}z^{4}+23a^{2}z^{2}+6a^{2}z^{-2}-19a^{2}+2az^{7}-8az^{5}+21az^{3}+3z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-23az-7za^{-1}+12az^{-1}+5a^{-1}z^{-1}+z^{6}-2z^{4}+7z^{2}+3z^{-2}-10$ (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 χ 4                 2 2 2               1 1 0 0             5 1   4 -2           3 5     2 -4         4 1       3 -6       2 5         3 -8     2 2           0 -10     2             2 -12 1 2               -1 -14 1                 1

Integral Khovanov Homology (db, data source)

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

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.

L10n108 L10n110