L11a90

L11a89

L11a91

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Link Presentations

[edit Notes on L11a90's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X18,8,19,7 X20,10,21,9 X22,15,5,16 X8,20,9,19 X16,21,17,22 X14,12,15,11 X10,18,11,17 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 4, -9, 8, -2, 11, -8, 5, -7, 9, -3, 6, -4, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-4t(2)^{3}+3t(2)^{2}-4t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{17/2}-4q^{15/2}+8q^{13/2}-12q^{11/2}+15q^{9/2}-17q^{7/2}+16q^{5/2}-13q^{3/2}+9{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+4z^{5}a^{-3}-2z^{5}a^{-5}+az^{3}-7z^{3}a^{-1}+7z^{3}a^{-3}-5z^{3}a^{-5}+z^{3}a^{-7}+3az-8za^{-1}+7za^{-3}-3za^{-5}+za^{-7}+2az^{-1}-4a^{-1}z^{-1}+3a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-2z^{9}a^{-1}-6z^{9}a^{-3}-4z^{9}a^{-5}-4z^{8}a^{-2}-10z^{8}a^{-4}-8z^{8}a^{-6}-2z^{8}-az^{7}+2z^{7}a^{-1}+8z^{7}a^{-3}-5z^{7}a^{-5}-10z^{7}a^{-7}+17z^{6}a^{-2}+25z^{6}a^{-4}+7z^{6}a^{-6}-8z^{6}a^{-8}+7z^{6}+5az^{5}+13z^{5}a^{-1}+14z^{5}a^{-3}+23z^{5}a^{-5}+13z^{5}a^{-7}-4z^{5}a^{-9}-9z^{4}a^{-2}-9z^{4}a^{-4}+3z^{4}a^{-6}+8z^{4}a^{-8}-z^{4}a^{-10}-6z^{4}-9az^{3}-26z^{3}a^{-1}-24z^{3}a^{-3}-15z^{3}a^{-5}-6z^{3}a^{-7}+2z^{3}a^{-9}-7z^{2}a^{-2}-6z^{2}a^{-4}-2z^{2}a^{-6}-2z^{2}a^{-8}-z^{2}+7az+17za^{-1}+13za^{-3}+4za^{-5}+za^{-7}+3a^{-2}+3a^{-4}+a^{-6}+2-2az^{-1}-4a^{-1}z^{-1}-3a^{-3}z^{-1}-a^{-5}z^{-1}$ (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 7 χ 18                       1 -1 16                     3   3 14                   5 1   -4 12                 7 3     4 10               8 5       -3 8             9 7         2 6           7 8           1 4         6 9             -3 2       5 9               4 0     1 4                 -3 -2   1 5                   4 -4   1                     -1 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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