L11a243

L11a242

L11a244

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Link Presentations

[edit Notes on L11a243's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X22,10,7,9 X16,11,17,12 X10,15,11,16 X2738 X20,17,21,18 X6,14,1,13 X4,20,5,19 X18,6,19,5 X14,21,15,22
Gauss code	{1, -6, 2, -9, 10, -8}, {6, -1, 3, -5, 4, -2, 8, -11, 5, -4, 7, -10, 9, -7, 11, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {\left(v^{2}-v+1\right)(uv-u-2v+1)(uv-2u-v+1)}{uv^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {9}{q^{9/2}}}-4q^{7/2}+{\frac {16}{q^{7/2}}}+9q^{5/2}-{\frac {21}{q^{5/2}}}-16q^{3/2}+{\frac {24}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+20{\sqrt {q}}-{\frac {25}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z-2a^{3}z^{5}-4a^{3}z^{3}+z^{3}a^{-3}-3a^{3}z+za^{-3}-a^{3}z^{-1}+az^{7}+3az^{5}-2z^{5}a^{-1}+5az^{3}-4z^{3}a^{-1}+5az-3za^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-5a^{6}z^{4}+2a^{6}z^{2}+8a^{5}z^{7}-11a^{5}z^{5}+6a^{5}z^{3}-a^{5}z+10a^{4}z^{8}-13a^{4}z^{6}+z^{6}a^{-4}+6a^{4}z^{4}-2z^{4}a^{-4}+z^{2}a^{-4}-a^{4}+7a^{3}z^{9}+a^{3}z^{7}+4z^{7}a^{-3}-16a^{3}z^{5}-9z^{5}a^{-3}+11a^{3}z^{3}+7z^{3}a^{-3}-2a^{3}z-2za^{-3}+a^{3}z^{-1}+2a^{2}z^{10}+18a^{2}z^{8}+7z^{8}a^{-2}-41a^{2}z^{6}-14z^{6}a^{-2}+26a^{2}z^{4}+7z^{4}a^{-2}-3a^{2}z^{2}-3a^{2}+13az^{9}+6z^{9}a^{-1}-14az^{7}-3z^{7}a^{-1}-10az^{5}-15z^{5}a^{-1}+12az^{3}+15z^{3}a^{-1}-5az-6za^{-1}+3az^{-1}+2a^{-1}z^{-1}+2z^{10}+15z^{8}-39z^{6}+24z^{4}-2z^{2}-3$ (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                     3   3 6                   6 1   -5 4                 10 3     7 2               11 7       -4 0             14 9         5 -2           11 12           1 -4         10 13             -3 -6       6 11               5 -8     3 10                 -7 -10   1 6                   5 -12   3                     -3 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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