L11a68

L11a67

L11a69

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Link Presentations

[edit Notes on L11a68's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(4v^{2}-5v+4\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{21/2}-3q^{19/2}+7q^{17/2}-11q^{15/2}+15q^{13/2}-17q^{11/2}+16q^{9/2}-15q^{7/2}+10q^{5/2}-6q^{3/2}+2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}+za^{-9}+a^{-9}z^{-1}-z^{5}a^{-7}-z^{3}a^{-7}-2za^{-7}-2a^{-7}z^{-1}-2z^{5}a^{-5}-3z^{3}a^{-5}+a^{-5}z^{-1}-z^{5}a^{-3}-z^{3}a^{-3}-za^{-3}-a^{-3}z^{-1}+z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-12}-3z^{4}a^{-12}+2z^{2}a^{-12}+3z^{7}a^{-11}-8z^{5}a^{-11}+5z^{3}a^{-11}+5z^{8}a^{-10}-14z^{6}a^{-10}+14z^{4}a^{-10}-9z^{2}a^{-10}+3a^{-10}+4z^{9}a^{-9}-7z^{7}a^{-9}+4z^{5}a^{-9}-6z^{3}a^{-9}+3za^{-9}-a^{-9}z^{-1}+z^{10}a^{-8}+9z^{8}a^{-8}-30z^{6}a^{-8}+37z^{4}a^{-8}-26z^{2}a^{-8}+7a^{-8}+7z^{9}a^{-7}-13z^{7}a^{-7}+12z^{5}a^{-7}-10z^{3}a^{-7}+6za^{-7}-2a^{-7}z^{-1}+z^{10}a^{-6}+8z^{8}a^{-6}-22z^{6}a^{-6}+28z^{4}a^{-6}-17z^{2}a^{-6}+4a^{-6}+3z^{9}a^{-5}-3z^{5}a^{-5}+z^{3}a^{-5}+3za^{-5}-a^{-5}z^{-1}+4z^{8}a^{-4}-5z^{6}a^{-4}+5z^{4}a^{-4}-2z^{2}a^{-4}+3z^{7}a^{-3}-2z^{5}a^{-3}-3z^{3}a^{-3}+3za^{-3}-a^{-3}z^{-1}+2z^{6}a^{-2}-3z^{4}a^{-2}+a^{-2}+z^{5}a^{-1}-3z^{3}a^{-1}+3za^{-1}-a^{-1}z^{-1}$ (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                 6 2     4 14               9 5       -4 12             8 6         2 10           8 9           1 8         7 8             -1 6       3 8               5 4     3 7                 -4 2   1 5                   4 0   1                     -1 -2 1                       1

Integral Khovanov Homology (db, data source)

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

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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