L11a267

L11a266

L11a268

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a267 at Knotilus!

Link Presentations

[edit Notes on L11a267's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{3}v^{3}-3u^{3}v^{2}+3u^{3}v-u^{3}-3u^{2}v^{3}+7u^{2}v^{2}-6u^{2}v+3u^{2}+3uv^{3}-6uv^{2}+7uv-3u-v^{3}+3v^{2}-3v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+7q^{7/2}-11q^{5/2}+16q^{3/2}-17{\sqrt {q}}+{\frac {16}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+2z^{3}a^{-3}-6a^{3}z-a^{3}z^{-1}-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+9az^{3}-6z^{3}a^{-1}+6az-3za^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-4a^{5}z^{5}+4z^{5}a^{-5}+6a^{5}z^{3}-3z^{3}a^{-5}-4a^{5}z+a^{5}z^{-1}+3a^{4}z^{8}-11a^{4}z^{6}+7z^{6}a^{-4}+12a^{4}z^{4}-7z^{4}a^{-4}-3a^{4}z^{2}+3z^{2}a^{-4}-a^{4}+3a^{3}z^{9}-5a^{3}z^{7}+8z^{7}a^{-3}-10a^{3}z^{5}-6z^{5}a^{-3}+21a^{3}z^{3}-z^{3}a^{-3}-10a^{3}z-za^{-3}+a^{3}z^{-1}+a^{2}z^{10}+8a^{2}z^{8}+7z^{8}a^{-2}-33a^{2}z^{6}-2z^{6}a^{-2}+30a^{2}z^{4}-14z^{4}a^{-2}-6a^{2}z^{2}+11z^{2}a^{-2}+7az^{9}+4z^{9}a^{-1}-8az^{7}+6z^{7}a^{-1}-24az^{5}-28z^{5}a^{-1}+36az^{3}+23z^{3}a^{-1}-13az-8za^{-1}+z^{10}+12z^{8}-31z^{6}+12z^{4}+5z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   4 1   -3 6                 7 3     4 4               9 4       -5 2             8 7         1 0           9 10           1 -2         6 7             -1 -4       4 9               5 -6     3 6                 -3 -8   1 5                   4 -10   2                     -2 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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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