L10n51

L10n50

L10n52

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Link Presentations

[edit Notes on L10n51's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X11,19,12,18 X5,12,6,13 X17,4,18,5 X14,7,15,8 X16,14,17,13 X20,15,7,16 X6,19,1,20
Gauss code	{1, -2, 3, 6, -5, -10}, {7, -1, 2, -3, -4, 5, 8, -7, 9, -8, -6, 4, 10, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-2u^{2}v^{2}+u^{2}v+uv^{4}-3uv^{3}+3uv^{2}-3uv+u+v^{3}-2v^{2}}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {6}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {4}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-2a^{7}z^{3}-4a^{7}z-2a^{7}z^{-1}+a^{5}z^{5}+3a^{5}z^{3}+4a^{5}z+2a^{5}z^{-1}-2a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-2z^{6}a^{10}+5z^{4}a^{10}-2z^{2}a^{10}-2z^{7}a^{9}+4z^{5}a^{9}-2z^{3}a^{9}+3za^{9}-a^{9}z^{-1}-z^{8}a^{8}+2z^{4}a^{8}-4z^{7}a^{7}+11z^{5}a^{7}-15z^{3}a^{7}+9za^{7}-2a^{7}z^{-1}-z^{8}a^{6}+z^{6}a^{6}-3z^{4}a^{6}+2z^{2}a^{6}-a^{6}-2z^{7}a^{5}+6z^{5}a^{5}-13z^{3}a^{5}+9za^{5}-2a^{5}z^{-1}-z^{6}a^{4}-3z^{3}a^{3}+5za^{3}-a^{3}z^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                 2 2 -4               2 1 -1 -6             3 1   2 -8           3 2     -1 -10         3 3       0 -12       2 3         1 -14     2 3           -1 -16   1 3             2 -18   1               -1 -20 1                 1

Integral Khovanov Homology (db, data source)

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

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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