L10n8

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Link Presentations

[edit Notes on L10n8's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X17,1,18,4 X9,14,10,15 X3849 X5,11,6,10 X11,5,12,20 X13,19,14,18 X19,13,20,12 X2,16,3,15
Gauss code	{1, -10, -5, 3}, {-6, -1, 2, 5, -4, 6, -7, 9, -8, 4, 10, -2, -3, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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