L10a86

L10a85

L10a87

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Link Presentations

[edit Notes on L10a86's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-3u^{2}v^{3}+4u^{2}v^{2}-3u^{2}v+u^{2}-2uv^{4}+7uv^{3}-11uv^{2}+7uv-2u+v^{4}-3v^{3}+4v^{2}-3v+1}{uv^{2}}}$ (db)
Jones polynomial	$-9q^{9/2}+13q^{7/2}-{\frac {1}{q^{7/2}}}-17q^{5/2}+{\frac {4}{q^{5/2}}}+18q^{3/2}-{\frac {9}{q^{3/2}}}-q^{13/2}+4q^{11/2}-17{\sqrt {q}}+{\frac {13}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}+2az^{3}-7z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+2az-5za^{-1}+5za^{-3}-za^{-5}+az^{-1}-2a^{-1}z^{-1}+2a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-3z^{9}a^{-1}-3z^{9}a^{-3}-16z^{8}a^{-2}-8z^{8}a^{-4}-8z^{8}-8az^{7}-12z^{7}a^{-1}-12z^{7}a^{-3}-8z^{7}a^{-5}-4a^{2}z^{6}+30z^{6}a^{-2}+11z^{6}a^{-4}-4z^{6}a^{-6}+11z^{6}-a^{3}z^{5}+14az^{5}+36z^{5}a^{-1}+36z^{5}a^{-3}+14z^{5}a^{-5}-z^{5}a^{-7}+5a^{2}z^{4}-18z^{4}a^{-2}-4z^{4}a^{-4}+5z^{4}a^{-6}-4z^{4}+a^{3}z^{3}-9az^{3}-31z^{3}a^{-1}-31z^{3}a^{-3}-9z^{3}a^{-5}+z^{3}a^{-7}-a^{2}z^{2}+4z^{2}a^{-2}+z^{2}a^{-4}-z^{2}a^{-6}+z^{2}+4az+12za^{-1}+12za^{-3}+4za^{-5}-a^{-2}-az^{-1}-2a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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 6 χ 14                     1 1 12                   3   -3 10                 6 1   5 8               7 3     -4 6             10 6       4 4           9 8         -1 2         8 9           -1 0       6 10             4 -2     3 7               -4 -4   1 6                 5 -6   3                   -3 -8 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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