L10n3

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n3 at Knotilus!

Link Presentations

[edit Notes on L10n3's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 X11,19,12,18 X17,5,18,20 X19,13,20,12 X9,16,10,17 X2,14,3,13
Gauss code	{1, -10, 5, -3}, {-4, -1, 2, -5, -9, 4, -6, 8, 10, -2, 3, 9, -7, 6, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}+q^{7/2}-{\frac {2}{q^{7/2}}}-2q^{5/2}+{\frac {3}{q^{5/2}}}+3q^{3/2}-{\frac {4}{q^{3/2}}}-5{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{3}-2a^{3}z+za^{-3}-a^{3}z^{-1}+a^{-3}z^{-1}+az^{5}+4az^{3}-2z^{3}a^{-1}+6az-5za^{-1}+4az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{4}z^{6}-4a^{4}z^{4}+4a^{4}z^{2}+z^{2}a^{-4}-a^{4}-a^{-4}+2a^{3}z^{7}-8a^{3}z^{5}+8a^{3}z^{3}+2z^{3}a^{-3}-4a^{3}z-2za^{-3}+a^{3}z^{-1}+a^{-3}z^{-1}+a^{2}z^{8}-a^{2}z^{6}+z^{6}a^{-2}-8a^{2}z^{4}-3z^{4}a^{-2}+11a^{2}z^{2}+7z^{2}a^{-2}-4a^{2}-4a^{-2}+4az^{7}+2z^{7}a^{-1}-16az^{5}-8z^{5}a^{-1}+19az^{3}+13z^{3}a^{-1}-11az-9za^{-1}+4az^{-1}+4a^{-1}z^{-1}+z^{8}-z^{6}-7z^{4}+13z^{2}-7$ (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 χ 8                 1 -1 6               1   1 4             2 1   -1 2           3 1     2 0         2 4       2 -2       2 2 1       1 -4     1 2           1 -6   1 2             -1 -8   1               1 -10 1                 -1

Integral Khovanov Homology (db, data source)

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

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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