L10n52

L10n51

L10n53

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Link Presentations

[edit Notes on L10n52's Link Presentations]

Planar diagram presentation	X8192 X9,19,10,18 X14,6,15,5 X16,12,17,11 X3,10,4,11 X12,7,13,8 X20,15,7,16 X6,14,1,13 X4,19,5,20 X17,2,18,3
Gauss code	{1, 10, -5, -9, 3, -8}, {6, -1, -2, 5, 4, -6, 8, -3, 7, -4, -10, 2, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-u^{2}v^{2}+2u^{2}v-u^{2}+uv^{4}-3uv^{3}+5uv^{2}-3uv+u-v^{4}+2v^{3}-v^{2}}{uv^{2}}}$ (db)
Jones polynomial	$-2q^{3/2}+4{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {8}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+za^{5}+a^{5}z^{-1}-z^{5}a^{3}-3z^{3}a^{3}-5za^{3}-2a^{3}z^{-1}+3z^{3}a+5za+2az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{8}-2a^{2}z^{8}-4a^{5}z^{7}-7a^{3}z^{7}-3az^{7}-3a^{6}z^{6}+2a^{4}z^{6}+4a^{2}z^{6}-z^{6}-a^{7}z^{5}+11a^{5}z^{5}+21a^{3}z^{5}+9az^{5}+7a^{6}z^{4}+2a^{4}z^{4}-7a^{2}z^{4}-2z^{4}+2a^{7}z^{3}-9a^{5}z^{3}-26a^{3}z^{3}-18az^{3}-3z^{3}a^{-1}-2a^{6}z^{2}-a^{4}z^{2}+4a^{2}z^{2}+3z^{2}+4a^{5}z+12a^{3}z+12az+4za^{-1}-a^{2}-a^{5}z^{-1}-2a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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 χ 4                 2 2 2               2   -2 0             4 2   2 -2           4 3     -1 -4         4 3       1 -6       3 5         2 -8     2 3           -1 -10   1 3             2 -12   2               -2 -14 1                 1

Integral Khovanov Homology (db, data source)

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

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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