L11a222

L11a221

L11a223

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Link Presentations

[edit Notes on L11a222's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X22,10,7,9 X20,12,21,11 X10,22,11,21 X16,6,17,5 X18,16,19,15 X14,20,15,19 X2738 X4,14,5,13 X6,18,1,17
Gauss code	{1, -9, 2, -10, 6, -11}, {9, -1, 3, -5, 4, -2, 10, -8, 7, -6, 11, -7, 8, -4, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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