L11a99

L11a98

L11a100

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Link Presentations

[edit Notes on L11a99's Link Presentations]

Planar diagram presentation	X6172 X2,9,3,10 X12,3,13,4 X10,5,11,6 X22,11,5,12 X4,21,1,22 X18,16,19,15 X16,8,17,7 X8,18,9,17 X20,14,21,13 X14,20,15,19
Gauss code	{1, -2, 3, -6}, {4, -1, 8, -9, 2, -4, 5, -3, 10, -11, 7, -8, 9, -7, 11, -10, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {4t(1)t(2)^{3}-3t(2)^{3}-10t(1)t(2)^{2}+10t(2)^{2}+10t(1)t(2)-10t(2)-3t(1)+4}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {6}{q^{9/2}}}-4q^{7/2}+{\frac {10}{q^{7/2}}}+8q^{5/2}-{\frac {15}{q^{5/2}}}-11q^{3/2}+{\frac {16}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {3}{q^{11/2}}}+15{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+za^{5}-z^{5}a^{3}-z^{3}a^{3}+2a^{3}z^{-1}-2z^{5}a-4z^{3}a-5za-3az^{-1}-z^{5}a^{-1}+2za^{-1}+a^{-1}z^{-1}+z^{3}a^{-3}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-4a^{3}z^{9}-10az^{9}-6z^{9}a^{-1}-4a^{4}z^{8}-2a^{2}z^{8}-7z^{8}a^{-2}-5z^{8}-4a^{5}z^{7}+6a^{3}z^{7}+30az^{7}+16z^{7}a^{-1}-4z^{7}a^{-3}-3a^{6}z^{6}+2a^{4}z^{6}+10a^{2}z^{6}+21z^{6}a^{-2}-z^{6}a^{-4}+27z^{6}-a^{7}z^{5}+5a^{5}z^{5}-9a^{3}z^{5}-38az^{5}-13z^{5}a^{-1}+10z^{5}a^{-3}+6a^{6}z^{4}+5a^{4}z^{4}-14a^{2}z^{4}-16z^{4}a^{-2}+2z^{4}a^{-4}-31z^{4}+2a^{7}z^{3}+9a^{3}z^{3}+20az^{3}+6z^{3}a^{-1}-3z^{3}a^{-3}-3a^{6}z^{2}-3a^{4}z^{2}+7a^{2}z^{2}+4z^{2}a^{-2}+11z^{2}-a^{7}z-4a^{3}z-7az-2za^{-1}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   5 1   -4 4                 6 3     3 2               9 5       -4 0             9 6         3 -2           8 10           2 -4         7 8             -1 -6       3 8               5 -8     3 7                 -4 -10   1 4                   3 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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