L11n130

L11n129

L11n131

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n130 at Knotilus!

Link Presentations

[edit Notes on L11n130's Link Presentations]

Planar diagram presentation	X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X5,13,6,12 X6718 X9,16,10,17 X13,20,14,21 X15,22,16,7 X19,4,20,5 X21,14,22,15
Gauss code	{1, 4, -3, 10, -5, -6}, {6, -1, -7, 3, -2, 5, -8, 11, -9, 7, -4, 2, -10, 8, -11, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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