L11a8

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Link Presentations

[edit Notes on L11a8's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X20,9,21,10 X8493 X18,22,19,21 X14,12,15,11 X12,5,13,6 X22,13,5,14 X10,19,11,20 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {8, -1, 2, -5, 4, -10, 7, -8, 9, -7, 11, -2, 3, -6, 10, -4, 6, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-5t(2)^{3}+9t(2)^{2}-5t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+5q^{5/2}-11q^{3/2}+17{\sqrt {q}}-{\frac {25}{\sqrt {q}}}+{\frac {27}{q^{3/2}}}-{\frac {27}{q^{5/2}}}+{\frac {23}{q^{7/2}}}-{\frac {17}{q^{9/2}}}+{\frac {10}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+3z^{3}a^{5}+2za^{5}-a^{5}z^{-1}-3z^{5}a^{3}-5z^{3}a^{3}+3a^{3}z^{-1}+z^{7}a+3z^{5}a+4z^{3}a-za-2az^{-1}-z^{5}a^{-1}-z^{3}a^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-8a^{7}z^{5}+6a^{7}z^{3}-2a^{7}z+8a^{6}z^{8}-17a^{6}z^{6}+14a^{6}z^{4}-6a^{6}z^{2}+a^{6}+7a^{5}z^{9}-3a^{5}z^{7}-18a^{5}z^{5}+18a^{5}z^{3}-4a^{5}z-a^{5}z^{-1}+2a^{4}z^{10}+22a^{4}z^{8}-59a^{4}z^{6}+47a^{4}z^{4}-16a^{4}z^{2}+3a^{4}+15a^{3}z^{9}-8a^{3}z^{7}-35a^{3}z^{5}+z^{5}a^{-3}+30a^{3}z^{3}-a^{3}z-3a^{3}z^{-1}+2a^{2}z^{10}+27a^{2}z^{8}-61a^{2}z^{6}+5z^{6}a^{-2}+39a^{2}z^{4}-4z^{4}a^{-2}-11a^{2}z^{2}+3a^{2}+8az^{9}+10az^{7}+11z^{7}a^{-1}-41az^{5}-15z^{5}a^{-1}+25az^{3}+7z^{3}a^{-1}+az-2az^{-1}+13z^{8}-15z^{6}+4z^{4}-2z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     4   -4 4                   7 1   6 2                 10 4     -6 0               15 7       8 -2             14 12         -2 -4           13 13           0 -6         10 14             4 -8       7 13               -6 -10     3 10                 7 -12   1 7                   -6 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{15}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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