L11n162

L11n161

L11n163

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Link Presentations

[edit Notes on L11n162's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X17,13,18,12 X14,5,15,6 X4,13,5,14 X11,19,12,18 X19,7,20,22 X15,21,16,20 X21,17,22,16 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, -6, 3, 5, -4, -8, 9, -3, 6, -7, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{4}+t(2)^{4}+t(1)^{2}t(2)^{3}-2t(1)t(2)^{3}-t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-t(2)^{2}-2t(1)t(2)+t(2)+t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+3q^{5/2}-{\frac {2}{q^{5/2}}}-4q^{3/2}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{11/2}}}+4{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$za^{5}+a^{5}z^{-1}-z^{5}a-4z^{3}a-5za-2az^{-1}-z^{5}a^{-1}-3z^{3}a^{-1}-za^{-1}+a^{-1}z^{-1}+z^{3}a^{-3}+2za^{-3}$ (db)
Kauffman polynomial	$a^{5}z^{7}-7a^{5}z^{5}+14a^{5}z^{3}-9a^{5}z+a^{5}z^{-1}+z^{6}a^{-4}-2a^{4}z^{4}-4z^{4}a^{-4}+4a^{4}z^{2}+3z^{2}a^{-4}-a^{4}+2z^{7}a^{-3}-2a^{3}z^{5}-8z^{5}a^{-3}+6a^{3}z^{3}+7z^{3}a^{-3}-3a^{3}z-2za^{-3}+a^{2}z^{8}+2z^{8}a^{-2}-7a^{2}z^{6}-8z^{6}a^{-2}+15a^{2}z^{4}+8z^{4}a^{-2}-10a^{2}z^{2}-5z^{2}a^{-2}+3a^{2}+2a^{-2}+az^{9}+z^{9}a^{-1}-6az^{7}-3z^{7}a^{-1}+13az^{5}-13az^{3}+2z^{3}a^{-1}+8az-2az^{-1}-a^{-1}z^{-1}+3z^{8}-16z^{6}+29z^{4}-22z^{2}+5$ (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 3 4 5 χ 10                       1 -1 8                     1   1 6                   2 1   -1 4                 2 1     1 2               2 2       0 0           1 3 2         0 -2           2 3           1 -4       1 2 2             -1 -6         2               2 -8     1 1                 0 -10 1                       1 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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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