L10n40

L10n39

L10n41

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n40 at Knotilus!

Link Presentations

[edit Notes on L10n40's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X15,5,16,4 X5,13,6,12 X11,16,12,17 X6,18,1,17 X14,19,15,20 X18,13,19,14
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 $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{4}-t(2)^{4}-2t(1)t(2)^{3}+t(2)^{3}+t(1)t(2)^{2}+t(1)^{2}t(2)-2t(1)t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+{\frac {1}{q^{7/2}}}+2q^{5/2}-{\frac {2}{q^{5/2}}}-4q^{3/2}+{\frac {3}{q^{3/2}}}+3{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{-3}-a^{3}z+2za^{-3}-a^{3}z^{-1}-z^{5}a^{-1}+2az^{3}-4z^{3}a^{-1}+5az-5za^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-4}-4z^{4}a^{-4}+a^{4}z^{2}+3z^{2}a^{-4}-a^{4}+2z^{7}a^{-3}-9z^{5}a^{-3}+2a^{3}z^{3}+11z^{3}a^{-3}-2a^{3}z-5za^{-3}+a^{3}z^{-1}+z^{8}a^{-2}+a^{2}z^{6}-2z^{6}a^{-2}-3a^{2}z^{4}-4z^{4}a^{-2}+6a^{2}z^{2}+5z^{2}a^{-2}-3a^{2}+2az^{7}+4z^{7}a^{-1}-9az^{5}-18z^{5}a^{-1}+16az^{3}+25z^{3}a^{-1}-11az-14za^{-1}+3az^{-1}+2a^{-1}z^{-1}+z^{8}-2z^{6}-3z^{4}+7z^{2}-3$ (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   \ -3 -2 -1 0 1 2 3 4 5 χ 10                 1 -1 8               1   1 6             1 1   0 4           3 1     2 2         1 2       1 0       3 2         1 -2     1 2           1 -4   1 2             -1 -6   1               1 -8 1                 -1

Integral Khovanov Homology (db, data source)

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

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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