L10n19

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L10n20

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n19 at Knotilus!

Link Presentations

[edit Notes on L10n19's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X11,14,12,15 X3849 X5,13,6,12 X13,5,14,20 X9,16,10,17 X15,10,16,11 X17,2,18,3
Gauss code	{1, 10, -5, -3}, {-6, -1, 2, 5, -8, 9, -4, 6, -7, 4, -9, 8, -10, -2, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(1)t(2)^{3}-4t(1)t(2)^{2}+t(2)^{2}+t(1)t(2)-4t(2)+2}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {5}{q^{9/2}}}-{\frac {5}{q^{7/2}}}+{\frac {4}{q^{5/2}}}-{\frac {3}{q^{3/2}}}-{\frac {2}{q^{15/2}}}+{\frac {3}{q^{13/2}}}-{\frac {4}{q^{11/2}}}-{\sqrt {q}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-a^{7}z^{3}-3a^{7}z-2a^{7}z^{-1}+a^{5}z^{5}+3a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}+a^{3}z^{5}+3a^{3}z^{3}+2a^{3}z+a^{3}z^{-1}-az^{3}-3az-az^{-1}$ (db)
Kauffman polynomial	$3a^{9}z^{3}-5a^{9}z+a^{9}z^{-1}+a^{8}z^{6}-2a^{8}z^{2}+a^{8}+2a^{7}z^{7}-8a^{7}z^{5}+18a^{7}z^{3}-13a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}-3a^{6}z^{6}+8a^{6}z^{4}-7a^{6}z^{2}+3a^{6}+3a^{5}z^{7}-10a^{5}z^{5}+16a^{5}z^{3}-9a^{5}z+a^{5}z^{-1}+a^{4}z^{8}-3a^{4}z^{6}+6a^{4}z^{4}-6a^{4}z^{2}+2a^{4}+a^{3}z^{7}-a^{3}z^{5}-3a^{3}z^{3}+3a^{3}z-a^{3}z^{-1}+a^{2}z^{6}-2a^{2}z^{4}-a^{2}z^{2}+a^{2}+az^{5}-4az^{3}+4az-az^{-1}$ (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 χ 2                 1 1 0                   0 -2             3 1   2 -4           2 1     -1 -6         3 2       1 -8       2 2         0 -10     2 3           -1 -12   2 3             1 -14   1               -1 -16 2                 2

Integral Khovanov Homology (db, data source)

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

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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