L11a96

L11a95

L11a97

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Link Presentations

[edit Notes on L11a96's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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