L11a224

L11a223

L11a225

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a224 at Knotilus!

Link Presentations

[edit Notes on L11a224's Link Presentations]

Planar diagram presentation	X8192 X16,6,17,5 X18,10,19,9 X10,20,11,19 X2,11,3,12 X12,3,13,4 X4758 X20,14,21,13 X22,16,7,15 X6,18,1,17 X14,22,15,21
Gauss code	{1, -5, 6, -7, 2, -10}, {7, -1, 3, -4, 5, -6, 8, -11, 9, -2, 10, -3, 4, -8, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{6}-2u^{2}v^{5}+3u^{2}v^{4}-3u^{2}v^{3}+2u^{2}v^{2}-u^{2}v-uv^{6}+4uv^{5}-6uv^{4}+7uv^{3}-6uv^{2}+4uv-u-v^{5}+2v^{4}-3v^{3}+3v^{2}-2v+1}{uv^{3}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+7q^{13/2}-11q^{11/2}+15q^{9/2}-17q^{7/2}+16q^{5/2}-15q^{3/2}+10{\sqrt {q}}-{\frac {7}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-5}+5z^{5}a^{-5}+9z^{3}a^{-5}+7za^{-5}+2a^{-5}z^{-1}-z^{9}a^{-3}-7z^{7}a^{-3}-19z^{5}a^{-3}-25z^{3}a^{-3}-16za^{-3}-5a^{-3}z^{-1}+z^{7}a^{-1}+5z^{5}a^{-1}+9z^{3}a^{-1}+8za^{-1}+3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}-z^{2}a^{-10}+3z^{5}a^{-9}-2z^{3}a^{-9}+6z^{6}a^{-8}-7z^{4}a^{-8}+5z^{2}a^{-8}-a^{-8}+8z^{7}a^{-7}-11z^{5}a^{-7}+7z^{3}a^{-7}+8z^{8}a^{-6}-12z^{6}a^{-6}+6z^{4}a^{-6}+6z^{9}a^{-5}-10z^{7}a^{-5}+10z^{5}a^{-5}-15z^{3}a^{-5}+9za^{-5}-2a^{-5}z^{-1}+2z^{10}a^{-4}+6z^{8}a^{-4}-26z^{6}a^{-4}+27z^{4}a^{-4}-18z^{2}a^{-4}+5a^{-4}+10z^{9}a^{-3}-32z^{7}a^{-3}+40z^{5}a^{-3}-37z^{3}a^{-3}+20za^{-3}-5a^{-3}z^{-1}+2z^{10}a^{-2}+z^{8}a^{-2}-19z^{6}a^{-2}+24z^{4}a^{-2}-14z^{2}a^{-2}+5a^{-2}+4z^{9}a^{-1}+az^{7}-13z^{7}a^{-1}-4az^{5}+12z^{5}a^{-1}+5az^{3}-8z^{3}a^{-1}-2az+9za^{-1}-3a^{-1}z^{-1}+3z^{8}-11z^{6}+11z^{4}-2z^{2}$ (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   \ -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               9 5       -4 8             8 6         2 6           8 9           1 4         7 8             -1 2       4 9               5 0     3 6                 -3 -2   1 5                   4 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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