L11n76

L11n75

L11n77

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Link Presentations

[edit Notes on L11n76's Link Presentations]

Planar diagram presentation	X6172 X3,10,4,11 X15,5,16,22 X7,17,8,16 X17,20,18,21 X9,14,10,15 X19,13,20,12 X13,19,14,18 X21,9,22,8 X2536 X11,4,12,1
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 9, -6, 2, -11, 7, -8, 6, -3, 4, -5, 8, -7, 5, -9, 3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{5}-5t(1)t(2)^{4}+t(2)^{4}+8t(1)t(2)^{3}-6t(2)^{3}-6t(1)t(2)^{2}+8t(2)^{2}+t(1)t(2)-5t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {3}{q^{9/2}}}-3q^{7/2}+{\frac {7}{q^{7/2}}}+7q^{5/2}-{\frac {11}{q^{5/2}}}-11q^{3/2}+{\frac {13}{q^{3/2}}}+13{\sqrt {q}}-{\frac {15}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}+z^{3}a^{-3}-4a^{3}z+2za^{-3}-2a^{3}z^{-1}+a^{-3}z^{-1}+az^{7}+4az^{5}-2z^{5}a^{-1}+6az^{3}-6z^{3}a^{-1}+5az-6za^{-1}+3az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2az^{9}-2z^{9}a^{-1}-7a^{2}z^{8}-4z^{8}a^{-2}-11z^{8}-8a^{3}z^{7}-11az^{7}-6z^{7}a^{-1}-3z^{7}a^{-3}-3a^{4}z^{6}+12a^{2}z^{6}+7z^{6}a^{-2}-z^{6}a^{-4}+23z^{6}+16a^{3}z^{5}+35az^{5}+27z^{5}a^{-1}+8z^{5}a^{-3}-3a^{4}z^{4}-13a^{2}z^{4}+2z^{4}a^{-2}+3z^{4}a^{-4}-11z^{4}-6a^{5}z^{3}-21a^{3}z^{3}-33az^{3}-25z^{3}a^{-1}-7z^{3}a^{-3}+3a^{4}z^{2}+9a^{2}z^{2}-6z^{2}a^{-2}-3z^{2}a^{-4}+3z^{2}+5a^{5}z+11a^{3}z+15az+12za^{-1}+3za^{-3}-2a^{2}+2a^{-2}+a^{-4}-a^{5}z^{-1}-2a^{3}z^{-1}-3az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               5 1   -4 4             6 2     4 2           7 5       -2 0         8 6         2 -2       6 8           2 -4     5 7             -2 -6   2 6               4 -8 1 5                 -4 -10 3                   3

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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