L11a219

L11a218

L11a220

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a219 at Knotilus!

Link Presentations

[edit Notes on L11a219's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X20,10,21,9 X22,12,7,11 X10,22,11,21 X2738 X18,15,19,16 X6,14,1,13 X4,18,5,17 X16,6,17,5 X14,19,15,20
Gauss code	{1, -6, 2, -9, 10, -8}, {6, -1, 3, -5, 4, -2, 8, -11, 7, -10, 9, -7, 11, -3, 5, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+2t(2)^{4}-3t(1)^{2}t(2)^{3}+7t(1)t(2)^{3}-3t(2)^{3}+3t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+3t(2)^{2}-3t(1)^{2}t(2)+7t(1)t(2)-3t(2)+2t(1)^{2}-3t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+7q^{13/2}-11q^{11/2}+14q^{9/2}-17q^{7/2}+15q^{5/2}-14q^{3/2}+10{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}+a^{-7}z^{-1}-2z^{5}a^{-5}-6z^{3}a^{-5}-6za^{-5}-3a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+7z^{3}a^{-3}+7za^{-3}+2a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+2az-4za^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-3z^{9}a^{-1}-7z^{9}a^{-3}-4z^{9}a^{-5}-7z^{8}a^{-2}-11z^{8}a^{-4}-7z^{8}a^{-6}-3z^{8}-az^{7}+7z^{7}a^{-1}+13z^{7}a^{-3}-3z^{7}a^{-5}-8z^{7}a^{-7}+32z^{6}a^{-2}+33z^{6}a^{-4}+7z^{6}a^{-6}-6z^{6}a^{-8}+12z^{6}+4az^{5}+2z^{5}a^{-1}+5z^{5}a^{-3}+22z^{5}a^{-5}+12z^{5}a^{-7}-3z^{5}a^{-9}-34z^{4}a^{-2}-24z^{4}a^{-4}+3z^{4}a^{-6}+7z^{4}a^{-8}-z^{4}a^{-10}-15z^{4}-5az^{3}-9z^{3}a^{-1}-12z^{3}a^{-3}-21z^{3}a^{-5}-11z^{3}a^{-7}+2z^{3}a^{-9}+11z^{2}a^{-2}+4z^{2}a^{-4}-7z^{2}a^{-6}-5z^{2}a^{-8}+z^{2}a^{-10}+6z^{2}+2az+2za^{-1}+5za^{-3}+10za^{-5}+5za^{-7}+3a^{-4}+3a^{-6}+a^{-8}-2a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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 6 7 χ 18                       1 -1 16                     2   2 14                   5 1   -4 12                 6 2     4 10               8 5       -3 8             9 6         3 6           7 9           2 4         7 8             -1 2       4 8               4 0     2 6                 -4 -2   1 4                   3 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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