L11a88

L11a87

L11a89

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Link Presentations

[edit Notes on L11a88's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X18,8,19,7 X22,20,5,19 X20,14,21,13 X14,22,15,21 X16,10,17,9 X10,16,11,15 X8,18,9,17 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 7, -8, 11, -2, 5, -6, 8, -7, 9, -3, 4, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv^{3}-7uv^{2}+9uv-5u-5v^{3}+9v^{2}-7v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-9q^{9/2}+13q^{7/2}-{\frac {1}{q^{7/2}}}-15q^{5/2}+{\frac {2}{q^{5/2}}}+14q^{3/2}-{\frac {6}{q^{3/2}}}+q^{15/2}-3q^{13/2}+6q^{11/2}-13{\sqrt {q}}+{\frac {9}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}-2z^{3}a^{-5}-za^{-5}+a^{-5}z^{-1}+z^{5}a^{-3}+a^{3}z-za^{-3}+a^{3}z^{-1}-2a^{-3}z^{-1}+z^{5}a^{-1}-2az^{3}-2az-az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-3z^{4}a^{-8}+z^{2}a^{-8}+3z^{7}a^{-7}-9z^{5}a^{-7}+5z^{3}a^{-7}-za^{-7}+5z^{8}a^{-6}-17z^{6}a^{-6}+18z^{4}a^{-6}-10z^{2}a^{-6}+3a^{-6}+4z^{9}a^{-5}-10z^{7}a^{-5}+5z^{5}a^{-5}+z^{3}a^{-5}+za^{-5}-a^{-5}z^{-1}+z^{10}a^{-4}+8z^{8}a^{-4}-38z^{6}a^{-4}+58z^{4}a^{-4}-34z^{2}a^{-4}+7a^{-4}+7z^{9}a^{-3}-21z^{7}a^{-3}+a^{3}z^{5}+30z^{5}a^{-3}-3a^{3}z^{3}-21z^{3}a^{-3}+3a^{3}z+9za^{-3}-a^{3}z^{-1}-2a^{-3}z^{-1}+z^{10}a^{-2}+6z^{8}a^{-2}+2a^{2}z^{6}-23z^{6}a^{-2}-3a^{2}z^{4}+37z^{4}a^{-2}-23z^{2}a^{-2}+a^{2}+4a^{-2}+3z^{9}a^{-1}+3az^{7}-5z^{7}a^{-1}-3az^{5}+12z^{5}a^{-1}-2az^{3}-16z^{3}a^{-1}+3az+7za^{-1}-az^{-1}-a^{-1}z^{-1}+3z^{8}-z^{6}-3z^{4}$ (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                 5 2     3 8               8 4       -4 6             7 5         2 4           7 8           1 2         6 7             -1 0       4 8               4 -2     2 5                 -3 -4     4                   4 -6 1 2                     -1 -8 1                       1

Integral Khovanov Homology (db, data source)

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

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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