L11a288

L11a287

L11a289

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a288 at Knotilus!

Link Presentations

[edit Notes on L11a288's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u+v)\left(uv^{2}-uv+u+v-1\right)\left(uv^{2}-uv-v^{2}+v-1\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$q^{15/2}-3q^{13/2}+6q^{11/2}-11q^{9/2}+13q^{7/2}-16q^{5/2}+16q^{3/2}-14{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-5}+3z^{3}a^{-5}+2za^{-5}-a^{-5}z^{-1}-z^{7}a^{-3}-4z^{5}a^{-3}-5z^{3}a^{-3}-za^{-3}+a^{-3}z^{-1}-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+3az^{3}-5z^{3}a^{-1}+2az-za^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}a^{-4}-5z^{9}a^{-1}-9z^{9}a^{-3}-4z^{9}a^{-5}-2z^{8}a^{-2}-4z^{8}a^{-6}-6z^{8}-5az^{7}+12z^{7}a^{-1}+29z^{7}a^{-3}+9z^{7}a^{-5}-3z^{7}a^{-7}-3a^{2}z^{6}+16z^{6}a^{-2}+8z^{6}a^{-4}+9z^{6}a^{-6}-z^{6}a^{-8}+15z^{6}-a^{3}z^{5}+11az^{5}-13z^{5}a^{-1}-40z^{5}a^{-3}-6z^{5}a^{-5}+9z^{5}a^{-7}+6a^{2}z^{4}-26z^{4}a^{-2}-9z^{4}a^{-4}-2z^{4}a^{-6}+3z^{4}a^{-8}-16z^{4}+2a^{3}z^{3}-7az^{3}+2z^{3}a^{-1}+21z^{3}a^{-3}+3z^{3}a^{-5}-7z^{3}a^{-7}-a^{2}z^{2}+9z^{2}a^{-2}+3z^{2}a^{-4}-z^{2}a^{-6}-2z^{2}a^{-8}+6z^{2}+2az-za^{-1}-2za^{-3}+3za^{-5}+2za^{-7}+a^{-4}-a^{-3}z^{-1}-a^{-5}z^{-1}$ (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 16                       1 -1 14                     2   2 12                   4 1   -3 10                 7 2     5 8               7 5       -2 6             9 6         3 4           7 7           0 2         7 9             -2 0       4 8               4 -2     2 6                 -4 -4   1 4                   3 -6   2                     -2 -8 1                       1

Integral Khovanov Homology (db, data source)

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

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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