L10n28

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L10n29

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Link Presentations

[edit Notes on L10n28's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X11,19,12,18 X15,20,16,5 X19,16,20,17 X17,13,18,12 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, -3, 8, 10, -2, -4, 7, -8, 3, -5, 6, -7, 4, -6, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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