L11a229

L11a228

L11a230

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a229 at Knotilus!

Link Presentations

[edit Notes on L11a229's Link Presentations]

Planar diagram presentation	X8192 X2,11,3,12 X12,3,13,4 X16,5,17,6 X6718 X4,15,5,16 X20,14,21,13 X18,10,19,9 X10,20,11,19 X22,18,7,17 X14,22,15,21
Gauss code	{1, -2, 3, -6, 4, -5}, {5, -1, 8, -9, 2, -3, 7, -11, 6, -4, 10, -8, 9, -7, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{6}-3u^{2}v^{5}+5u^{2}v^{4}-5u^{2}v^{3}+3u^{2}v^{2}+3uv^{5}-7uv^{4}+9uv^{3}-7uv^{2}+3uv+3v^{4}-5v^{3}+5v^{2}-3v+1}{uv^{3}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+8q^{5/2}-13q^{3/2}+17{\sqrt {q}}-{\frac {21}{\sqrt {q}}}+{\frac {20}{q^{3/2}}}-{\frac {18}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{9}+a^{3}z^{7}-7az^{7}+z^{7}a^{-1}+5a^{3}z^{5}-20az^{5}+5z^{5}a^{-1}+10a^{3}z^{3}-29az^{3}+10z^{3}a^{-1}+9a^{3}z-21az+9za^{-1}+3a^{3}z^{-1}-5az^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+a^{7}z+3a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+6a^{5}z^{7}-10a^{5}z^{5}+8a^{5}z^{3}-3a^{5}z+7a^{4}z^{8}-9a^{4}z^{6}+z^{6}a^{-4}+4a^{4}z^{4}-3z^{4}a^{-4}+3z^{2}a^{-4}-a^{-4}+6a^{3}z^{9}-8a^{3}z^{7}+3z^{7}a^{-3}+12a^{3}z^{5}-7z^{5}a^{-3}-17a^{3}z^{3}+4z^{3}a^{-3}+12a^{3}z-3a^{3}z^{-1}+2a^{2}z^{10}+9a^{2}z^{8}+5z^{8}a^{-2}-25a^{2}z^{6}-10z^{6}a^{-2}+28a^{2}z^{4}+4z^{4}a^{-2}-18a^{2}z^{2}+5a^{2}+11az^{9}+5z^{9}a^{-1}-25az^{7}-8z^{7}a^{-1}+35az^{5}+5z^{5}a^{-1}-41az^{3}-10z^{3}a^{-1}+25az+9za^{-1}-5az^{-1}-2a^{-1}z^{-1}+2z^{10}+7z^{8}-24z^{6}+27z^{4}-20z^{2}+5$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     2   2 6                   6 1   -5 4                 7 2     5 2               10 6       -4 0             11 7         4 -2           10 11           1 -4         8 10             -2 -6       5 10               5 -8     3 8                 -5 -10     5                   5 -12 1 3                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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