L11a93

L11a92

L11a94

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Link Presentations

[edit Notes on L11a93's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{4}-4v^{3}+3v^{2}-4v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+7q^{13/2}-12q^{11/2}+15q^{9/2}-17q^{7/2}+16q^{5/2}-14q^{3/2}+10{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}+a^{-7}z^{-1}-2z^{5}a^{-5}-6z^{3}a^{-5}-7za^{-5}-4a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+8z^{3}a^{-3}+11za^{-3}+6a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-7z^{3}a^{-1}+3az-9za^{-1}+2az^{-1}-5a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-2z^{9}a^{-1}-6z^{9}a^{-3}-4z^{9}a^{-5}-5z^{8}a^{-2}-11z^{8}a^{-4}-8z^{8}a^{-6}-2z^{8}-az^{7}+z^{7}a^{-1}+4z^{7}a^{-3}-7z^{7}a^{-5}-9z^{7}a^{-7}+20z^{6}a^{-2}+28z^{6}a^{-4}+9z^{6}a^{-6}-6z^{6}a^{-8}+7z^{6}+5az^{5}+17z^{5}a^{-1}+31z^{5}a^{-3}+37z^{5}a^{-5}+15z^{5}a^{-7}-3z^{5}a^{-9}-13z^{4}a^{-2}-10z^{4}a^{-4}+3z^{4}a^{-6}+6z^{4}a^{-8}-z^{4}a^{-10}-7z^{4}-9az^{3}-33z^{3}a^{-1}-48z^{3}a^{-3}-41z^{3}a^{-5}-15z^{3}a^{-7}+2z^{3}a^{-9}-z^{2}a^{-2}-6z^{2}a^{-4}-9z^{2}a^{-6}-4z^{2}a^{-8}+z^{2}a^{-10}+z^{2}+7az+22za^{-1}+29za^{-3}+20za^{-5}+6za^{-7}+a^{-2}+3a^{-4}+3a^{-6}+a^{-8}+1-2az^{-1}-5a^{-1}z^{-1}-6a^{-3}z^{-1}-4a^{-5}z^{-1}-a^{-7}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     2   2 14                   5 1   -4 12                 7 2     5 10               8 5       -3 8             9 7         2 6           7 8           1 4         7 9             -2 2       5 9               4 0     1 5                 -4 -2   1 5                   4 -4   1                     -1 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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