L11n11

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n11 at Knotilus!

Link Presentations

[edit Notes on L11n11's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
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	$-{\frac {2}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {6}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {4}{q^{17/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)-2a^{9}z-a^{9}z^{-1}+a^{7}z^{5}+3a^{7}z^{3}+3a^{7}z+a^{7}z^{-1}+a^{5}z^{5}+3a^{5}z^{3}+4a^{5}z+2a^{5}z^{-1}-2a^{3}z^{3}-5a^{3}z-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{6}-4a^{12}z^{4}+4a^{12}z^{2}-a^{12}+2a^{11}z^{7}-7a^{11}z^{5}+5a^{11}z^{3}-a^{11}z+2a^{10}z^{8}-6a^{10}z^{6}+3a^{10}z^{4}-a^{10}z^{2}+a^{9}z^{9}-a^{9}z^{7}-4a^{9}z^{5}+3a^{9}z^{3}-a^{9}z^{-1}+4a^{8}z^{8}-15a^{8}z^{6}+20a^{8}z^{4}-12a^{8}z^{2}+3a^{8}+a^{7}z^{9}-a^{7}z^{7}-4a^{7}z^{5}+8a^{7}z^{3}-a^{7}z^{-1}+2a^{6}z^{8}-7a^{6}z^{6}+11a^{6}z^{4}-3a^{6}z^{2}+2a^{5}z^{7}-7a^{5}z^{5}+13a^{5}z^{3}-8a^{5}z+2a^{5}z^{-1}+a^{4}z^{6}-2a^{4}z^{4}+4a^{4}z^{2}-3a^{4}+3a^{3}z^{3}-7a^{3}z+2a^{3}z^{-1}$ (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                 2 2 0 -6               3     3 -8             3 2     -1 -10           4 3       1 -12         2 3         1 -14       3 4           -1 -16     1 2             1 -18   1 3               -2 -20   1                 1 -22 1                   -1

Integral Khovanov Homology (db, data source)

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

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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