L10n5

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n5 at Knotilus!

Link Presentations

[edit Notes on L10n5's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$2q^{9/2}-2q^{7/2}+2q^{5/2}-{\frac {1}{q^{5/2}}}-3q^{3/2}+{\frac {1}{q^{3/2}}}-q^{11/2}+2{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-1}+az^{3}-5z^{3}a^{-1}+2z^{3}a^{-3}+3az-8za^{-1}+6za^{-3}-za^{-5}+2az^{-1}-4a^{-1}z^{-1}+3a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-2}-z^{8}-az^{7}-3z^{7}a^{-1}-2z^{7}a^{-3}+3z^{6}a^{-2}-2z^{6}a^{-4}+5z^{6}+6az^{5}+16z^{5}a^{-1}+9z^{5}a^{-3}-z^{5}a^{-5}+2z^{4}a^{-2}+8z^{4}a^{-4}-6z^{4}-11az^{3}-26z^{3}a^{-1}-13z^{3}a^{-3}+2z^{3}a^{-5}-7z^{2}a^{-2}-9z^{2}a^{-4}-2z^{2}a^{-6}+7az+17za^{-1}+11za^{-3}-za^{-7}+3a^{-2}+3a^{-4}+a^{-6}+2-2az^{-1}-4a^{-1}z^{-1}-3a^{-3}z^{-1}-a^{-5}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 χ 12                 1 1 10               1   -1 8             2 2   0 6           1 1     0 4         2 2 1     1 2       2 3         1 0       1 1         0 -2   1 2             1 -4                   0 -6 1                 1

Integral Khovanov Homology (db, data source)

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

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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