L10n86

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L10n87

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Link Presentations

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Planar diagram presentation	X6172 X10,3,11,4 X13,20,14,15 X16,8,17,7 X8,16,9,15 X11,18,12,19 X19,12,20,13 X17,14,18,5 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {5, -4, -8, 6, -7, 3}, {9, -1, 4, -5, 10, -2, -6, 7, -3, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(3)-1)\left(t(1)t(3)t(2)^{2}-t(3)t(2)^{2}-t(1)t(3)t(2)-t(2)-t(1)+1\right)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$1-2q^{-1}+3q^{-2}-3q^{-3}+5q^{-4}-3q^{-5}+4q^{-6}-2q^{-7}+q^{-8}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}+z^{4}a^{6}+2z^{2}a^{6}-2a^{6}z^{-2}-2a^{6}-z^{6}a^{4}-4z^{4}a^{4}-3z^{2}a^{4}+a^{4}z^{-2}+a^{4}+z^{4}a^{2}+3z^{2}a^{2}+a^{2}$ (db)
Kauffman polynomial	$z^{2}a^{10}+2z^{3}a^{9}+4z^{4}a^{8}-6z^{2}a^{8}-a^{8}z^{-2}+4a^{8}+z^{7}a^{7}-2z^{5}a^{7}+4z^{3}a^{7}-5za^{7}+2a^{7}z^{-1}+z^{8}a^{6}-3z^{6}a^{6}+6z^{4}a^{6}-10z^{2}a^{6}-2a^{6}z^{-2}+6a^{6}+3z^{7}a^{5}-10z^{5}a^{5}+9z^{3}a^{5}-5za^{5}+2a^{5}z^{-1}+z^{8}a^{4}-2z^{6}a^{4}-2z^{4}a^{4}+z^{2}a^{4}-a^{4}z^{-2}+2a^{4}+2z^{7}a^{3}-8z^{5}a^{3}+7z^{3}a^{3}+z^{6}a^{2}-4z^{4}a^{2}+4z^{2}a^{2}-a^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 1                 1 1 -1               1   -1 -3             2 1   1 -5           2 2     0 -7         3 1       2 -9       1 3         2 -11     3 2           1 -13     2             2 -15 1 2               -1 -17 1                 1

Integral Khovanov Homology (db, data source)

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

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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