L11a256

L11a255

L11a257

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a256 at Knotilus!

Link Presentations

[edit Notes on L11a256's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{3}v^{4}-2u^{3}v^{3}+2u^{3}v^{2}-u^{3}v+u^{2}v^{5}-4u^{2}v^{4}+7u^{2}v^{3}-8u^{2}v^{2}+4u^{2}v-u^{2}-uv^{5}+4uv^{4}-8uv^{3}+7uv^{2}-4uv+u-v^{4}+2v^{3}-2v^{2}+v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {12}{q^{9/2}}}-q^{7/2}+{\frac {16}{q^{7/2}}}+4q^{5/2}-{\frac {20}{q^{5/2}}}-9q^{3/2}+{\frac {20}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {6}{q^{11/2}}}+14{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+4a^{3}z^{5}+3az^{5}-z^{5}a^{-1}-3a^{5}z^{3}+6a^{3}z^{3}+2az^{3}-2z^{3}a^{-1}-2a^{5}z+2a^{3}z-az-za^{-1}+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-4a^{5}z^{9}-10a^{3}z^{9}-6az^{9}-4a^{6}z^{8}-3a^{4}z^{8}-8a^{2}z^{8}-9z^{8}-3a^{7}z^{7}+6a^{5}z^{7}+21a^{3}z^{7}+4az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+8a^{6}z^{6}+9a^{4}z^{6}+18a^{2}z^{6}-4z^{6}a^{-2}+14z^{6}+9a^{7}z^{5}-a^{5}z^{5}-22a^{3}z^{5}+2az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}-2a^{6}z^{4}-4a^{4}z^{4}-11a^{2}z^{4}+5z^{4}a^{-2}-7z^{4}-8a^{7}z^{3}-3a^{5}z^{3}+13a^{3}z^{3}+az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}-2a^{8}z^{2}-2a^{6}z^{2}-a^{4}z^{2}+2a^{2}z^{2}-z^{2}a^{-2}+2z^{2}+2a^{7}z+4a^{5}z-a^{3}z-2az+za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 8 3     -5 0               10 6       4 -2             11 9         -2 -4           9 9           0 -6         7 11             4 -8       5 9               -4 -10     2 8                 6 -12   1 4                   -3 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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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