L11n86

L11n85

L11n87

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n86 at Knotilus!

Link Presentations

[edit Notes on L11n86's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X14,8,15,7 X18,10,19,9 X19,5,20,22 X15,21,16,20 X21,17,22,16 X8,18,9,17 X10,14,11,13 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -8, 4, -9, 11, -2, 9, -3, -6, 7, 8, -4, -5, 6, -7, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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