L11a94

L11a93

L11a95

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Link Presentations

[edit Notes on L11a94's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X20,8,21,7 X22,17,5,18 X18,21,19,22 X16,10,17,9 X14,12,15,11 X10,16,11,15 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, 8, -6, 4, -5, 9, -3, 5, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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