L11n330

L11n329

L11n331

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Link Presentations

[edit Notes on L11n330's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X13,18,14,19 X9,13,10,22 X11,21,12,20 X19,5,20,12 X21,11,22,10 X17,1,18,4
Gauss code	{1, -4, -3, 11}, {-2, -1, 5, 3, -7, 10, -8, 9}, {-6, 2, 4, -5, -11, 6, -9, 8, -10, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{2}-2uv^{2}w+uvw^{3}-2uvw^{2}+uvw-uw^{3}+uw^{2}+v^{3}(-w)+v^{3}-v^{2}w^{2}+2v^{2}w-v^{2}+2vw^{2}-vw}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$2q^{4}-2q^{3}+4q^{2}-5q+7-5q^{-1}+5q^{-2}-3q^{-3}+2q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}\left(-z^{2}\right)+a^{-4}z^{-2}-a^{4}+2a^{-4}+a^{2}z^{4}+2a^{2}z^{2}-3z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}-5a^{-2}+z^{4}+z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$a^{5}z^{5}-3a^{5}z^{3}+a^{5}z+2a^{4}z^{6}-6a^{4}z^{4}+3z^{4}a^{-4}+3a^{4}z^{2}-10z^{2}a^{-4}-a^{-4}z^{-2}-a^{4}+6a^{-4}+2a^{3}z^{7}+z^{7}a^{-3}-5a^{3}z^{5}-3z^{5}a^{-3}+a^{3}z^{3}+4z^{3}a^{-3}+a^{3}z-5za^{-3}+2a^{-3}z^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}-7a^{2}z^{6}-11z^{6}a^{-2}+9a^{2}z^{4}+27z^{4}a^{-2}-4a^{2}z^{2}-30z^{2}a^{-2}-2a^{-2}z^{-2}+13a^{-2}+az^{9}+z^{9}a^{-1}-2az^{7}-3z^{7}a^{-1}-2az^{5}+z^{5}a^{-1}+9az^{3}+9z^{3}a^{-1}-3az-8za^{-1}+2a^{-1}z^{-1}+4z^{8}-20z^{6}+39z^{4}-27z^{2}-z^{-2}+9$ (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 4 χ 9                   2 2 7                 1 1 0 5               3 1   2 3             2 1     -1 1           5 3       2 -1         3 5         2 -3       2 2           0 -5     1 3             2 -7   1 2               -1 -9   1                 1 -11 1                   -1

Integral Khovanov Homology (db, data source)

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

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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