L10n50

L10n49

L10n51

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Link Presentations

[edit Notes on L10n50's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X13,20,14,7 X12,5,13,6 X10,4,11,3 X4,15,5,16 X16,12,17,11 X19,14,20,15 X2,18,3,17
Gauss code	{1, -10, 6, -7, 5, -3}, {3, -1, 2, -6, 8, -5, -4, 9, 7, -8, 10, -2, -9, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-2t(1)^{2}t(2)^{3}+2t(1)t(2)^{3}+t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(2)^{2}+2t(1)t(2)-2t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+a^{5}z^{5}+4a^{5}z^{3}+3a^{5}z-a^{3}z^{7}-5a^{3}z^{5}-7a^{3}z^{3}-3a^{3}z+a^{3}z^{-1}+az^{5}+3az^{3}-az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{2}+3a^{7}z^{3}-2a^{7}z+a^{6}z^{6}+3a^{5}z^{7}-11a^{5}z^{5}+15a^{5}z^{3}-6a^{5}z+2a^{4}z^{8}-6a^{4}z^{6}+5a^{4}z^{4}-2a^{4}z^{2}+6a^{3}z^{7}-23a^{3}z^{5}+23a^{3}z^{3}-5a^{3}z-a^{3}z^{-1}+2a^{2}z^{8}-6a^{2}z^{6}+2a^{2}z^{4}+a^{2}+3az^{7}-12az^{5}+11az^{3}-az-az^{-1}+z^{6}-3z^{4}+z^{2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 χ 4                 1 -1 2               2   2 0             1 1   0 -2           4 2     2 -4         2 2       0 -6       3 3         0 -8     2 3           1 -10   1 2             -1 -12   2               2 -14 1                 -1

Integral Khovanov Homology (db, data source)

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

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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