L10n43

L10n42

L10n44

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n43 at Knotilus!

Link Presentations

[edit Notes on L10n43's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X4,15,5,16 X5,13,6,12 X16,12,17,11 X17,6,18,1 X19,15,20,14 X13,19,14,18
Gauss code	{1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, 7, 6, -10, 9, 5, -7, -8, 10, -9, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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