L11a114

L11a113

L11a115

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Link Presentations

[edit Notes on L11a114's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {4(u-1)(v-1)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$15q^{9/2}-15q^{7/2}+10q^{5/2}-7q^{3/2}+q^{21/2}-2q^{19/2}+5q^{17/2}-9q^{15/2}+13q^{13/2}-15q^{11/2}+3{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}+2za^{-9}+a^{-9}z^{-1}-z^{5}a^{-7}-2z^{3}a^{-7}-3za^{-7}-a^{-7}z^{-1}-2z^{5}a^{-5}-4z^{3}a^{-5}-3za^{-5}-2a^{-5}z^{-1}-z^{5}a^{-3}+3za^{-3}+2a^{-3}z^{-1}+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-12}-4z^{4}a^{-12}+4z^{2}a^{-12}+2z^{7}a^{-11}-6z^{5}a^{-11}+4z^{3}a^{-11}+3z^{8}a^{-10}-9z^{6}a^{-10}+12z^{4}a^{-10}-12z^{2}a^{-10}+4a^{-10}+3z^{9}a^{-9}-8z^{7}a^{-9}+12z^{5}a^{-9}-11z^{3}a^{-9}+2za^{-9}-a^{-9}z^{-1}+z^{10}a^{-8}+5z^{8}a^{-8}-22z^{6}a^{-8}+39z^{4}a^{-8}-34z^{2}a^{-8}+9a^{-8}+6z^{9}a^{-7}-12z^{7}a^{-7}+8z^{5}a^{-7}+3z^{3}a^{-7}-za^{-7}-a^{-7}z^{-1}+z^{10}a^{-6}+7z^{8}a^{-6}-21z^{6}a^{-6}+27z^{4}a^{-6}-15z^{2}a^{-6}+4a^{-6}+3z^{9}a^{-5}+3z^{7}a^{-5}-20z^{5}a^{-5}+27z^{3}a^{-5}-11za^{-5}+2a^{-5}z^{-1}+5z^{8}a^{-4}-6z^{6}a^{-4}-z^{4}a^{-4}+4z^{2}a^{-4}-2a^{-4}+5z^{7}a^{-3}-9z^{5}a^{-3}+7z^{3}a^{-3}-7za^{-3}+2a^{-3}z^{-1}+3z^{6}a^{-2}-5z^{4}a^{-2}+z^{2}a^{-2}+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$ (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                     1   1 18                   4 1   -3 16                 5 1     4 14               8 4       -4 12             7 5         2 10           8 8           0 8         7 7             0 6       3 8               5 4     4 7                 -3 2   1 5                   4 0   2                     -2 -2 1                       1

Integral Khovanov Homology (db, data source)

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

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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