L11n186

L11n185

L11n187

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n186 at Knotilus!

Link Presentations

[edit Notes on L11n186's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X18,10,19,9 X19,22,20,7 X13,20,14,21 X21,14,22,15 X10,16,11,15 X16,6,17,5 X2738 X4,11,5,12 X6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 3, -7, 10, -2, -5, 6, 7, -8, 11, -3, -4, 5, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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