L10n9

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Link Presentations

[edit Notes on L10n9's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X8493 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 $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$-q^{11/2}+q^{9/2}-q^{7/2}+q^{5/2}-q^{3/2}+{\sqrt {q}}-{\frac {1}{\sqrt {q}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{-5}-a^{-5}z^{-1}+z^{3}a^{-3}+a^{3}z^{-1}+3za^{-3}+2a^{-3}z^{-1}-az-az^{-1}-za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-2}-z^{8}a^{-4}-z^{7}a^{-1}-2z^{7}a^{-3}-z^{7}a^{-5}+6z^{6}a^{-2}+6z^{6}a^{-4}+6z^{5}a^{-1}+12z^{5}a^{-3}+6z^{5}a^{-5}-11z^{4}a^{-2}-10z^{4}a^{-4}-z^{4}-11z^{3}a^{-1}-21z^{3}a^{-3}-10z^{3}a^{-5}+9z^{2}a^{-2}+5z^{2}a^{-4}+4z^{2}-a^{3}z-az+8za^{-1}+13za^{-3}+5za^{-5}-a^{2}-3a^{-2}-a^{-4}-2+a^{3}z^{-1}+az^{-1}-a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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 χ 12                 1 1 10                   0 8             1 1   0 6         1 1       0 4         1 1       0 2     1 2 1         0 0   1 2 1           0 -2   1 2             1 -4 1                 1 -6 1                 1

Integral Khovanov Homology (db, data source)

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

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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