L11n20

L11n19

L11n21

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n20 at Knotilus!

Link Presentations

[edit Notes on L11n20's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X17,1,18,4 X5,12,6,13 X3849 X13,22,14,5 X21,14,22,15 X9,18,10,19 X11,20,12,21 X19,10,20,11 X2,16,3,15
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -8, 10, -9, 4, -6, 7, 11, -2, -3, 8, -10, 9, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {6}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)-3a^{9}z-2a^{9}z^{-1}+a^{7}z^{5}+4a^{7}z^{3}+7a^{7}z+4a^{7}z^{-1}+a^{5}z^{5}+2a^{5}z^{3}-a^{5}z^{-1}-2a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{6}-5a^{12}z^{4}+7a^{12}z^{2}-2a^{12}+a^{11}z^{7}-3a^{11}z^{5}+a^{11}z^{3}+a^{11}z+a^{10}z^{8}-2a^{10}z^{6}-a^{10}z^{4}+2a^{10}z^{2}-a^{10}+a^{9}z^{9}-4a^{9}z^{7}+11a^{9}z^{5}-17a^{9}z^{3}+8a^{9}z-2a^{9}z^{-1}+3a^{8}z^{8}-11a^{8}z^{6}+21a^{8}z^{4}-21a^{8}z^{2}+6a^{8}+a^{7}z^{9}-3a^{7}z^{7}+10a^{7}z^{5}-16a^{7}z^{3}+13a^{7}z-4a^{7}z^{-1}+2a^{6}z^{8}-7a^{6}z^{6}+17a^{6}z^{4}-15a^{6}z^{2}+5a^{6}+2a^{5}z^{7}-4a^{5}z^{5}+5a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+a^{4}z^{6}+a^{4}z^{2}-a^{4}+3a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   2 2 -4                 3 2 -1 -6               3     3 -8             3 3     0 -10           4 3       1 -12         2 3         1 -14       3 4           -1 -16       2             2 -18   1 3               -2 -20                     0 -22 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{3}$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }^{2}$

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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