L11n450

L11n449

L11n451

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Link Presentations

[edit Notes on L11n450's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X11,20,12,21 X13,19,14,22 X21,18,22,9 X17,13,18,12 X15,8,16,5 X7,14,8,15 X19,17,20,16 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -8, 7}, {-9, 3, -5, 4}, {-11, 2, -3, 6, -4, 8, -7, 9, -6, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(w-1)(x-1)^{2}(uvx+u(-v)+u-vx+x-1)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+7q^{5/2}-13q^{3/2}+14{\sqrt {q}}-{\frac {18}{\sqrt {q}}}+{\frac {14}{q^{3/2}}}-{\frac {14}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {4}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-a^{3}z^{5}+4az^{5}-2z^{5}a^{-1}-3a^{3}z^{3}+8az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}+a^{5}z-7a^{3}z+12az-8za^{-1}+2za^{-3}+2a^{5}z^{-1}-8a^{3}z^{-1}+11az^{-1}-6a^{-1}z^{-1}+a^{-3}z^{-1}+a^{5}z^{-3}-3a^{3}z^{-3}+3az^{-3}-a^{-1}z^{-3}$ (db)
Kauffman polynomial	$10a^{5}z^{3}-a^{5}z^{-3}-11a^{5}z+5a^{5}z^{-1}+6a^{4}z^{6}+z^{6}a^{-4}-4a^{4}z^{4}-3z^{4}a^{-4}+9a^{4}z^{2}+3z^{2}a^{-4}+3a^{4}z^{-2}-9a^{4}-a^{-4}+13a^{3}z^{7}+3z^{7}a^{-3}-31a^{3}z^{5}-8z^{5}a^{-3}+44a^{3}z^{3}+9z^{3}a^{-3}-3a^{3}z^{-3}-34a^{3}z-6za^{-3}+14a^{3}z^{-1}+2a^{-3}z^{-1}+9a^{2}z^{8}+4z^{8}a^{-2}-9a^{2}z^{6}-5z^{6}a^{-2}-9a^{2}z^{4}-7z^{4}a^{-2}+24a^{2}z^{2}+14z^{2}a^{-2}+6a^{2}z^{-2}-21a^{2}-6a^{-2}+2az^{9}+2z^{9}a^{-1}+20az^{7}+10z^{7}a^{-1}-62az^{5}-39z^{5}a^{-1}+67az^{3}+42z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-44az-27za^{-1}+18az^{-1}+11a^{-1}z^{-1}+13z^{8}-21z^{6}-9z^{4}+26z^{2}+3z^{-2}-18$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               5 1   -4 4             8 2     6 2           6 5       -1 0         12 8         4 -2       8 12           4 -4     6 6             0 -6   2 8               6 -8 2 6                 -4 -10 4                   4

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb {Z} }^{4}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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