L11n219

L11n218

L11n220

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n219 at Knotilus!

Link Presentations

[edit Notes on L11n219's Link Presentations]

Planar diagram presentation	X10,1,11,2 X8,9,1,10 X3,12,4,13 X22,16,9,15 X17,3,18,2 X4,22,5,21 X14,5,15,6 X13,21,14,20 X16,12,17,11 X19,7,20,6 X7,19,8,18
Gauss code	{1, 5, -3, -6, 7, 10, -11, -2}, {2, -1, 9, 3, -8, -7, 4, -9, -5, 11, -10, 8, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)\left(t(1)^{2}+t(1)+1\right)(t(2)-1)}{t(1)^{3/2}{\sqrt {t(2)}}}}$ (db)
Jones polynomial	$q^{13/2}-q^{11/2}+q^{9/2}-2q^{7/2}+q^{5/2}-q^{3/2}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{-5}+3za^{-5}+a^{-5}z^{-1}-z^{5}a^{-3}-5z^{3}a^{-3}-7za^{-3}-3a^{-3}z^{-1}+z^{3}a^{-1}+4za^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{8}a^{-6}-7z^{6}a^{-6}+15z^{4}a^{-6}-10z^{2}a^{-6}+a^{-6}+z^{9}a^{-5}-7z^{7}a^{-5}+16z^{5}a^{-5}-15z^{3}a^{-5}+6za^{-5}-a^{-5}z^{-1}+2z^{8}a^{-4}-13z^{6}a^{-4}+25z^{4}a^{-4}-16z^{2}a^{-4}+3a^{-4}+z^{9}a^{-3}-7z^{7}a^{-3}+17z^{5}a^{-3}-21z^{3}a^{-3}+13za^{-3}-3a^{-3}z^{-1}+z^{8}a^{-2}-6z^{6}a^{-2}+10z^{4}a^{-2}-7z^{2}a^{-2}+3a^{-2}+z^{5}a^{-1}-6z^{3}a^{-1}+7za^{-1}-2a^{-1}z^{-1}-z^{2}$ (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   \ 0 1 2 3 4 5 6 7 χ 14               1 -1 12                 0 10           1 1   0 8         1       1 6         1       1 4     1 1         0 2 1               1 0 2 1             1 -2 1               1

Integral Khovanov Homology (db, data source)

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

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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