L10n12

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Link Presentations

[edit Notes on L10n12's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X8493 X9,16,10,17 X17,20,18,5 X11,19,12,18 X19,11,20,10 X2,14,3,13
Gauss code	{1, -10, 5, -3}, {-4, -1, 2, -5, -6, 9, -8, 4, 10, -2, 3, 6, -7, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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