L11n191

L11n190

L11n192

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Link Presentations

[edit Notes on L11n191's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X16,6,17,5 X15,21,16,20 X21,19,22,18 X19,15,20,14 X2738 X4,11,5,12 X6,18,1,17
Gauss code	{1, -9, 2, -10, 5, -11}, {9, -1, 3, -4, 10, -2, 4, 8, -6, -5, 11, 7, -8, 6, -7, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-2t(1)^{2}t(2)^{3}+2t(1)t(2)^{3}-t(2)^{3}+t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(2)^{2}-t(1)^{2}t(2)+2t(1)t(2)-2t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$5q^{9/2}-6q^{7/2}+7q^{5/2}-{\frac {1}{q^{5/2}}}-7q^{3/2}+{\frac {2}{q^{3/2}}}+q^{13/2}-3q^{11/2}+5{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$za^{-7}-z^{5}a^{-5}-4z^{3}a^{-5}-3za^{-5}+z^{7}a^{-3}+5z^{5}a^{-3}+8z^{3}a^{-3}+6za^{-3}+a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-8z^{3}a^{-1}+3az-8za^{-1}+2az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{2}a^{-8}+3z^{3}a^{-7}-za^{-7}+z^{6}a^{-6}+z^{4}a^{-6}-z^{2}a^{-6}+3z^{7}a^{-5}-8z^{5}a^{-5}+8z^{3}a^{-5}-3za^{-5}+3z^{8}a^{-4}-8z^{6}a^{-4}+4z^{4}a^{-4}-a^{-4}+z^{9}a^{-3}+3z^{7}a^{-3}-21z^{5}a^{-3}+23z^{3}a^{-3}-8za^{-3}+a^{-3}z^{-1}+5z^{8}a^{-2}-17z^{6}a^{-2}+10z^{4}a^{-2}+4z^{2}a^{-2}-3a^{-2}+z^{9}a^{-1}+az^{7}+z^{7}a^{-1}-5az^{5}-18z^{5}a^{-1}+9az^{3}+27z^{3}a^{-1}-7az-13za^{-1}+2az^{-1}+3a^{-1}z^{-1}+2z^{8}-8z^{6}+7z^{4}+2z^{2}-3$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 14                   1 -1 12                 2   2 10               3 1   -2 8             3 2     1 6           4 3       -1 4         3 3         0 2       3 5           2 0     2 2             0 -2   1 4               3 -4   1                 -1 -6 1                   1

Integral Khovanov Homology (db, data source)

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

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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