L10a91

L10a90

L10a92

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Link Presentations

[edit Notes on L10a91's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X20,12,9,11 X2,9,3,10 X4,20,5,19 X14,6,15,5 X16,7,17,8 X18,16,19,15 X6,17,7,18 X8,14,1,13
Gauss code	{1, -4, 2, -5, 6, -9, 7, -10}, {4, -1, 3, -2, 10, -6, 8, -7, 9, -8, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(u^{2}v-u^{2}+uv^{2}-4uv+u-v^{2}+v\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-7q^{9/2}+10q^{7/2}-13q^{5/2}+13q^{3/2}-13{\sqrt {q}}+{\frac {9}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}+z^{5}a^{-3}+z^{3}a^{-3}+a^{3}z+za^{-3}+z^{5}a^{-1}-2az^{3}-az-za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-7z^{4}a^{-6}+2z^{2}a^{-6}+6z^{7}a^{-5}-11z^{5}a^{-5}+5z^{3}a^{-5}-za^{-5}+4z^{8}a^{-4}-11z^{4}a^{-4}+7z^{2}a^{-4}+z^{9}a^{-3}+11z^{7}a^{-3}+a^{3}z^{5}-26z^{5}a^{-3}-2a^{3}z^{3}+20z^{3}a^{-3}+a^{3}z-6za^{-3}+7z^{8}a^{-2}+3a^{2}z^{6}-6z^{6}a^{-2}-6a^{2}z^{4}-6z^{4}a^{-2}+3a^{2}z^{2}+7z^{2}a^{-2}+z^{9}a^{-1}+4az^{7}+9z^{7}a^{-1}-6az^{5}-21z^{5}a^{-1}+3az^{3}+19z^{3}a^{-1}-2az-8za^{-1}+az^{-1}+a^{-1}z^{-1}+3z^{8}+z^{6}-8z^{4}+5z^{2}-1$ (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                 4 1   3 8               6 3     -3 6             7 4       3 4           6 6         0 2         7 7           0 0       4 8             4 -2     2 5               -3 -4   1 4                 3 -6   2                   -2 -8 1                     1

Integral Khovanov Homology (db, data source)

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

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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