L11a39

L11a38

L11a40

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Link Presentations

[edit Notes on L11a39's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(3v^{2}-8v+3\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-13q^{9/2}+16q^{7/2}-{\frac {1}{q^{7/2}}}-18q^{5/2}+{\frac {2}{q^{5/2}}}+18q^{3/2}-{\frac {6}{q^{3/2}}}+q^{15/2}-4q^{13/2}+8q^{11/2}-15{\sqrt {q}}+{\frac {10}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}-3z^{3}a^{-5}-3za^{-5}-a^{-5}z^{-1}+2z^{5}a^{-3}+4z^{3}a^{-3}+a^{3}z+5za^{-3}+a^{3}z^{-1}+2a^{-3}z^{-1}+z^{5}a^{-1}-2az^{3}-z^{3}a^{-1}-2az-2za^{-1}-az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-2z^{4}a^{-8}+z^{2}a^{-8}+4z^{7}a^{-7}-10z^{5}a^{-7}+7z^{3}a^{-7}-2za^{-7}+6z^{8}a^{-6}-13z^{6}a^{-6}+5z^{4}a^{-6}-z^{2}a^{-6}+4z^{9}a^{-5}+z^{7}a^{-5}-24z^{5}a^{-5}+24z^{3}a^{-5}-9za^{-5}+a^{-5}z^{-1}+z^{10}a^{-4}+12z^{8}a^{-4}-34z^{6}a^{-4}+26z^{4}a^{-4}-7z^{2}a^{-4}+a^{-4}+7z^{9}a^{-3}-4z^{7}a^{-3}+a^{3}z^{5}-23z^{5}a^{-3}-3a^{3}z^{3}+36z^{3}a^{-3}+3a^{3}z-17za^{-3}-a^{3}z^{-1}+2a^{-3}z^{-1}+z^{10}a^{-2}+10z^{8}a^{-2}+2a^{2}z^{6}-27z^{6}a^{-2}-3a^{2}z^{4}+29z^{4}a^{-2}-12z^{2}a^{-2}+a^{2}+3a^{-2}+3z^{9}a^{-1}+3az^{7}+2z^{7}a^{-1}-2az^{5}-12z^{5}a^{-1}-3az^{3}+19z^{3}a^{-1}+3az-10za^{-1}-az^{-1}+a^{-1}z^{-1}+4z^{8}-5z^{6}+7z^{4}-7z^{2}+2$ (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 16                       1 -1 14                     3   3 12                   5 1   -4 10                 8 3     5 8               8 5       -3 6             10 8         2 4           8 8           0 2         7 10             -3 0       5 10               5 -2     1 5                 -4 -4   1 5                   4 -6   1                     -1 -8 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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