L11n166

L11n165

L11n167

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Link Presentations

[edit Notes on L11n166's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X11,19,12,18 X19,7,20,22 X15,21,16,20 X21,17,22,16 X17,13,18,12 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, -5, 9, 11, -4, -7, 8, -9, 5, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}+t(1)t(2)^{4}-t(1)^{2}t(2)^{3}-t(1)t(2)^{3}+t(2)^{3}+t(1)t(2)^{2}+t(1)^{2}t(2)-t(1)t(2)-t(2)+t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{7/2}-q^{5/2}+q^{3/2}-{\sqrt {q}}-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {2}{q^{5/2}}}+{\frac {2}{q^{7/2}}}-{\frac {2}{q^{9/2}}}+{\frac {1}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{7}-7az^{5}+z^{5}a^{-1}+a^{5}z^{3}+a^{3}z^{3}-15az^{3}+5z^{3}a^{-1}+2a^{5}z+a^{3}z-12az+6za^{-1}+a^{5}z^{-1}-2az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-a^{2}z^{8}-z^{8}a^{-2}-2z^{8}-a^{5}z^{7}+8az^{7}+7z^{7}a^{-1}-a^{6}z^{6}-a^{4}z^{6}+8a^{2}z^{6}+7z^{6}a^{-2}+15z^{6}-a^{7}z^{5}+4a^{5}z^{5}-20az^{5}-15z^{5}a^{-1}+3a^{6}z^{4}+3a^{4}z^{4}-18a^{2}z^{4}-15z^{4}a^{-2}-33z^{4}+4a^{7}z^{3}-5a^{5}z^{3}+22az^{3}+13z^{3}a^{-1}-a^{6}z^{2}-2a^{4}z^{2}+13a^{2}z^{2}+11z^{2}a^{-2}+25z^{2}-3a^{7}z+4a^{5}z-13az-6za^{-1}+a^{4}-3a^{2}-2a^{-2}-5-a^{5}z^{-1}+2az^{-1}+a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 8                       1 -1 6                         0 4                   1 1   0 2               1 1       0 0             1   1       2 -2           2 3 1         0 -4         1               1 -6       1 2 1             0 -8     1 1                 0 -10     1                   1 -12 1 1                     0 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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