L11a290

L11a289

L11a291

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a290 at Knotilus!

Link Presentations

[edit Notes on L11a290's Link Presentations]

Planar diagram presentation	X10,1,11,2 X20,13,21,14 X12,4,13,3 X2,19,3,20 X14,7,15,8 X16,5,17,6 X6,15,7,16 X8,9,1,10 X18,12,19,11 X22,18,9,17 X4,22,5,21
Gauss code	{1, -4, 3, -11, 6, -7, 5, -8}, {8, -1, 9, -3, 2, -5, 7, -6, 10, -9, 4, -2, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{3}v^{4}-3u^{3}v^{3}+3u^{3}v^{2}-u^{3}v+u^{2}v^{5}-4u^{2}v^{4}+8u^{2}v^{3}-9u^{2}v^{2}+4u^{2}v-u^{2}-uv^{5}+4uv^{4}-9uv^{3}+8uv^{2}-4uv+u-v^{4}+3v^{3}-3v^{2}+v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+15{\sqrt {q}}-{\frac {20}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {23}{q^{5/2}}}+{\frac {19}{q^{7/2}}}-{\frac {14}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+3a^{3}z^{5}+3az^{5}-z^{5}a^{-1}-2a^{5}z^{3}+2a^{3}z^{3}+3az^{3}-2z^{3}a^{-1}-2a^{3}z+az-za^{-1}+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+4a^{7}z^{7}-10a^{7}z^{5}+5a^{7}z^{3}+7a^{6}z^{8}-18a^{6}z^{6}+13a^{6}z^{4}-3a^{6}z^{2}+7a^{5}z^{9}-15a^{5}z^{7}+9a^{5}z^{5}-a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+3a^{4}z^{10}+6a^{4}z^{8}-29a^{4}z^{6}+30a^{4}z^{4}-8a^{4}z^{2}-a^{4}+15a^{3}z^{9}-37a^{3}z^{7}+36a^{3}z^{5}+z^{5}a^{-3}-13a^{3}z^{3}-z^{3}a^{-3}-a^{3}z+a^{3}z^{-1}+3a^{2}z^{10}+9a^{2}z^{8}-30a^{2}z^{6}+4z^{6}a^{-2}+29a^{2}z^{4}-5z^{4}a^{-2}-9a^{2}z^{2}+z^{2}a^{-2}+8az^{9}-10az^{7}+8z^{7}a^{-1}+4az^{5}-12z^{5}a^{-1}-az^{3}+5z^{3}a^{-1}-za^{-1}+10z^{8}-16z^{6}+9z^{4}-3z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 9 3     -6 0               11 6       5 -2             12 10         -2 -4           11 10           1 -6         8 12             4 -8       6 11               -5 -10     3 9                 6 -12   1 5                   -4 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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