L10n90

L10n89

L10n91

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Link Presentations

[edit Notes on L10n90's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X13,20,14,17 X7,18,8,19 X17,10,18,11 X9,15,10,14 X15,9,16,8 X19,16,20,5 X2536 X11,1,12,4
Gauss code	{1, -9, -2, 10}, {-5, 4, -8, 3}, {9, -1, -4, 7, -6, 5, -10, 2, -3, 6, -7, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(3)-1)\left(t(2)t(3)^{2}-2t(2)t(3)+2t(3)-1\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{3}-3q^{2}+5q-7+9q^{-1}-7q^{-2}+8q^{-3}-5q^{-4}+3q^{-5}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-z^{4}a^{4}-3z^{2}a^{4}-2a^{4}z^{-2}-5a^{4}+z^{6}a^{2}+4z^{4}a^{2}+7z^{2}a^{2}+a^{2}z^{-2}+6a^{2}-2z^{4}-5z^{2}-3+z^{2}a^{-2}+a^{-2}$ (db)
Kauffman polynomial	$2a^{2}z^{8}+2z^{8}+6a^{3}z^{7}+9az^{7}+3z^{7}a^{-1}+7a^{4}z^{6}+5a^{2}z^{6}+z^{6}a^{-2}-z^{6}+3a^{5}z^{5}-12a^{3}z^{5}-25az^{5}-10z^{5}a^{-1}-15a^{4}z^{4}-23a^{2}z^{4}-3z^{4}a^{-2}-11z^{4}+7a^{3}z^{3}+16az^{3}+9z^{3}a^{-1}+6a^{6}z^{2}+16a^{4}z^{2}+18a^{2}z^{2}+3z^{2}a^{-2}+11z^{2}+3a^{5}z+a^{3}z-3az-za^{-1}-4a^{6}-8a^{4}-7a^{2}-a^{-2}-3-2a^{5}z^{-1}-2a^{3}z^{-1}+a^{6}z^{-2}+2a^{4}z^{-2}+a^{2}z^{-2}$ (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 7                 1 1 5               2   -2 3             3 1   2 1           4 2     -2 -1         5 3       2 -3       4 6         2 -5     4 3           1 -7   1 4             3 -9 2 4               -2 -11 3                 3

Integral Khovanov Homology (db, data source)

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

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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