L10a84

L10a83

L10a85

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Link Presentations

[edit Notes on L10a84's Link Presentations]

Planar diagram presentation	X8192 X14,9,15,10 X4758 X16,6,17,5 X18,16,19,15 X6,18,1,17 X20,11,7,12 X10,19,11,20 X2,14,3,13 X12,4,13,3
Gauss code	{1, -9, 10, -3, 4, -6}, {3, -1, 2, -8, 7, -10, 9, -2, 5, -4, 6, -5, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-2t(1)^{2}t(2)^{3}+6t(1)t(2)^{3}-3t(2)^{3}+3t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+3t(2)^{2}-3t(1)^{2}t(2)+6t(1)t(2)-2t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {3}{q^{9/2}}}-4q^{7/2}+{\frac {7}{q^{7/2}}}+7q^{5/2}-{\frac {11}{q^{5/2}}}-11q^{3/2}+{\frac {13}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+13{\sqrt {q}}-{\frac {15}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-3a^{3}z^{3}+z^{3}a^{-3}-4a^{3}z+za^{-3}-a^{3}z^{-1}+az^{7}+5az^{5}-2z^{5}a^{-1}+11az^{3}-6z^{3}a^{-1}+10az-6za^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-a^{6}z^{2}+3a^{5}z^{5}-2a^{5}z^{3}+6a^{4}z^{6}+z^{6}a^{-4}-7a^{4}z^{4}-2z^{4}a^{-4}+5a^{4}z^{2}+z^{2}a^{-4}-a^{4}+8a^{3}z^{7}+4z^{7}a^{-3}-13a^{3}z^{5}-11z^{5}a^{-3}+13a^{3}z^{3}+8z^{3}a^{-3}-6a^{3}z-za^{-3}+a^{3}z^{-1}+6a^{2}z^{8}+5z^{8}a^{-2}-4a^{2}z^{6}-11z^{6}a^{-2}-7a^{2}z^{4}+3z^{4}a^{-2}+10a^{2}z^{2}+2z^{2}a^{-2}-3a^{2}+2az^{9}+2z^{9}a^{-1}+11az^{7}+7z^{7}a^{-1}-35az^{5}-30z^{5}a^{-1}+33az^{3}+26z^{3}a^{-1}-15az-10za^{-1}+3az^{-1}+2a^{-1}z^{-1}+11z^{8}-22z^{6}+6z^{4}+5z^{2}-3$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                     1 -1 8                   3   3 6                 4 1   -3 4               7 3     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   2                   2 -12 1                     -1

Integral Khovanov Homology (db, data source)

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

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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