L11n126

L11n125

L11n127

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Link Presentations

[edit Notes on L11n126's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X15,5,16,4 X5,13,6,12 X11,16,12,17 X6,18,1,17 X14,20,15,19 X20,14,21,13 X18,21,19,22
Gauss code	{1, -4, 2, 5, -6, -8}, {4, -1, 3, -2, -7, 6, 10, -9, -5, 7, 8, -11, 9, -10, 11, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{4}-t(2)^{4}-4t(1)t(2)^{3}+3t(2)^{3}-2t(1)^{2}t(2)^{2}+7t(1)t(2)^{2}-2t(2)^{2}+3t(1)^{2}t(2)-4t(1)t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{15/2}-3q^{13/2}+5q^{11/2}-8q^{9/2}+9q^{7/2}-10q^{5/2}+9q^{3/2}-7{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {2}{q^{3/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-3z^{3}a^{-1}+2z^{3}a^{-3}-2z^{3}a^{-5}+2az-4za^{-1}+4za^{-3}-2za^{-5}+za^{-7}+az^{-1}-2a^{-1}z^{-1}+2a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-3}-z^{9}a^{-5}-3z^{8}a^{-2}-6z^{8}a^{-4}-3z^{8}a^{-6}-3z^{7}a^{-1}-5z^{7}a^{-3}-5z^{7}a^{-5}-3z^{7}a^{-7}+6z^{6}a^{-2}+15z^{6}a^{-4}+7z^{6}a^{-6}-z^{6}a^{-8}-z^{6}+7z^{5}a^{-1}+20z^{5}a^{-3}+23z^{5}a^{-5}+10z^{5}a^{-7}-8z^{4}a^{-2}-10z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}a^{-8}-2z^{4}-3az^{3}-15z^{3}a^{-1}-26z^{3}a^{-3}-23z^{3}a^{-5}-9z^{3}a^{-7}+3z^{2}a^{-2}+2z^{2}a^{-4}-z^{2}a^{-6}-2z^{2}a^{-8}+2z^{2}+4az+11za^{-1}+13za^{-3}+9za^{-5}+3za^{-7}-a^{-2}-az^{-1}-2a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 16                   1 -1 14                 2   2 12               3 1   -2 10             5 2     3 8           4 3       -1 6         6 5         1 4       4 5           1 2     3 5             -2 0   2 5               3 -2   2                 -2 -4 2                   2

Integral Khovanov Homology (db, data source)

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

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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