L11n96

L11n95

L11n97

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n96 at Knotilus!

Link Presentations

[edit Notes on L11n96's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,16,8,17 X9,18,10,19 X17,8,18,9 X22,20,5,19 X20,14,21,13 X14,22,15,21 X15,10,16,11 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 5, -4, 9, 11, -2, 7, -8, -9, 3, -5, 4, 6, -7, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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