L11n188

L11n187

L11n189

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Link Presentations

[edit Notes on L11n188's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X16,6,17,5 X22,13,7,14 X18,15,19,16 X14,21,15,22 X9,20,10,21 X4,18,5,17 X19,10,20,11 X2738 X6,11,1,12
Gauss code	{1, -10, 2, -8, 3, -11}, {10, -1, -7, 9, 11, -2, 4, -6, 5, -3, 8, -5, -9, 7, 6, -4}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{4}-t(2)^{4}+2t(1)^{2}t(2)^{3}-5t(1)t(2)^{3}+2t(2)^{3}-3t(1)^{2}t(2)^{2}+7t(1)t(2)^{2}-3t(2)^{2}+2t(1)^{2}t(2)-5t(1)t(2)+2t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {12}{q^{7/2}}}+{\frac {12}{q^{9/2}}}-{\frac {11}{q^{11/2}}}+{\frac {8}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {2}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-2a^{7}z^{3}-3a^{7}z-a^{7}z^{-1}+2a^{5}z^{5}+6a^{5}z^{3}+7a^{5}z+3a^{5}z^{-1}+a^{3}z^{5}-4a^{3}z-2a^{3}z^{-1}-az^{3}-az$ (db)
Kauffman polynomial	$3a^{10}z^{4}-4a^{10}z^{2}+a^{9}z^{7}+3a^{9}z^{5}-6a^{9}z^{3}+a^{9}z+2a^{8}z^{8}-2a^{8}z^{4}+2a^{8}z^{2}-a^{8}+a^{7}z^{9}+5a^{7}z^{7}-12a^{7}z^{5}+12a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}+6a^{6}z^{8}-7a^{6}z^{6}-3a^{6}z^{4}+10a^{6}z^{2}-3a^{6}+a^{5}z^{9}+9a^{5}z^{7}-26a^{5}z^{5}+28a^{5}z^{3}-14a^{5}z+3a^{5}z^{-1}+4a^{4}z^{8}-4a^{4}z^{6}-3a^{4}z^{4}+5a^{4}z^{2}-3a^{4}+5a^{3}z^{7}-10a^{3}z^{5}+8a^{3}z^{3}-7a^{3}z+2a^{3}z^{-1}+3a^{2}z^{6}-5a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az$ (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 χ 2                   1 1 0                 2   -2 -2               5 1   4 -4             5 3     -2 -6           7 4       3 -8         6 6         0 -10       5 6           -1 -12     3 6             3 -14   2 5               -3 -16   3                 3 -18 2                   -2

Integral Khovanov Homology (db, data source)

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

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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