L11n120

L11n119

L11n121

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n120 at Knotilus!

Link Presentations

[edit Notes on L11n120's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X9,16,10,17 X11,21,12,20 X21,9,22,8 X7,19,8,18 X19,13,20,12 X15,10,16,11 X17,5,18,22 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, -6, 5, -3, 8, -4, 7, 11, -2, -8, 3, -9, 6, -7, 4, -5, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{5}-t(1)t(2)^{4}-3t(2)^{4}+2t(1)t(2)^{3}+2t(2)^{2}-3t(1)t(2)-t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+2q^{5/2}-q^{3/2}-{\frac {2}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-2a^{3}z^{3}+z^{3}a^{-3}-5a^{3}z-2a^{3}z^{-1}+2za^{-3}+a^{-3}z^{-1}+az^{5}-z^{5}a^{-1}+5az^{3}-5z^{3}a^{-1}+7az+3az^{-1}-7za^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{5}z^{7}-5a^{5}z^{5}+7a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+2a^{4}z^{8}-10a^{4}z^{6}+z^{6}a^{-4}+12a^{4}z^{4}-4z^{4}a^{-4}-4a^{4}z^{2}+3z^{2}a^{-4}-a^{-4}+a^{3}z^{9}-3a^{3}z^{7}+2z^{7}a^{-3}-6a^{3}z^{5}-9z^{5}a^{-3}+16a^{3}z^{3}+8z^{3}a^{-3}-10a^{3}z-3za^{-3}+2a^{3}z^{-1}+a^{-3}z^{-1}+3a^{2}z^{8}+z^{8}a^{-2}-18a^{2}z^{6}-4z^{6}a^{-2}+27a^{2}z^{4}-z^{4}a^{-2}-13a^{2}z^{2}+6z^{2}a^{-2}+2a^{2}-2a^{-2}+az^{9}-4az^{7}+2z^{7}a^{-1}-6az^{5}-14z^{5}a^{-1}+23az^{3}+22z^{3}a^{-1}-16az-13za^{-1}+3az^{-1}+3a^{-1}z^{-1}+2z^{8}-13z^{6}+18z^{4}-6z^{2}$ (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 3 4 5 χ 10                       1 -1 8                     1   1 6                   1 1   0 4               2 2 1     -1 2             1 1 1       1 0           2 4 2         0 -2         2 2 2           2 -4       1 2 1             0 -6     2 2 1               1 -8   1 2                   1 -10   1                     -1 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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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