L11a65

L11a64

L11a66

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Link Presentations

[edit Notes on L11a65's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(4t(2)^{2}-5t(2)+4\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$16q^{9/2}-15q^{7/2}+10q^{5/2}-7q^{3/2}+q^{21/2}-3q^{19/2}+7q^{17/2}-10q^{15/2}+14q^{13/2}-17q^{11/2}+3{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}+za^{-9}+a^{-9}z^{-1}-z^{5}a^{-7}-z^{3}a^{-7}-za^{-7}-a^{-7}z^{-1}-2z^{5}a^{-5}-4z^{3}a^{-5}-4za^{-5}-2a^{-5}z^{-1}-z^{5}a^{-3}+3za^{-3}+2a^{-3}z^{-1}+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-3z^{9}a^{-5}-7z^{9}a^{-7}-4z^{9}a^{-9}-5z^{8}a^{-4}-9z^{8}a^{-6}-9z^{8}a^{-8}-5z^{8}a^{-10}-5z^{7}a^{-3}-4z^{7}a^{-5}+12z^{7}a^{-7}+8z^{7}a^{-9}-3z^{7}a^{-11}-3z^{6}a^{-2}+6z^{6}a^{-4}+27z^{6}a^{-6}+34z^{6}a^{-8}+15z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}+9z^{5}a^{-3}+22z^{5}a^{-5}-4z^{5}a^{-9}+8z^{5}a^{-11}+5z^{4}a^{-2}-32z^{4}a^{-6}-46z^{4}a^{-8}-16z^{4}a^{-10}+3z^{4}a^{-12}+2z^{3}a^{-1}-7z^{3}a^{-3}-29z^{3}a^{-5}-14z^{3}a^{-7}+2z^{3}a^{-9}-4z^{3}a^{-11}-z^{2}a^{-2}-3z^{2}a^{-4}+15z^{2}a^{-6}+31z^{2}a^{-8}+12z^{2}a^{-10}-2z^{2}a^{-12}-za^{-1}+7za^{-3}+13za^{-5}+5za^{-7}+2a^{-4}-4a^{-6}-9a^{-8}-4a^{-10}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-7}z^{-1}+a^{-9}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     2   2 18                   5 1   -4 16                 5 2     3 14               9 5       -4 12             8 5         3 10           8 9           1 8         7 8             -1 6       3 8               5 4     4 7                 -3 2   1 5                   4 0   2                     -2 -2 1                       1

Integral Khovanov Homology (db, data source)

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

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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