L11n136

L11n135

L11n137

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Link Presentations

[edit Notes on L11n136's Link Presentations]

Planar diagram presentation	X8192 X11,19,12,18 X3,10,4,11 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X20,14,21,13 X22,16,7,15 X19,4,20,5 X14,22,15,21
Gauss code	{1, -4, -3, 10, 5, -6}, {6, -1, 7, 3, -2, -5, 8, -11, 9, -7, 4, 2, -10, -8, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{6}-t(1)t(2)^{5}+t(2)^{5}+t(1)t(2)^{4}-t(1)t(2)^{3}+t(1)t(2)^{2}+t(1)^{2}t(2)-t(1)t(2)+t(1)}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}+q^{7/2}-{\frac {2}{q^{7/2}}}-q^{5/2}+{\frac {2}{q^{5/2}}}+2q^{3/2}-{\frac {3}{q^{3/2}}}-{\frac {1}{q^{11/2}}}-3{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{3}z^{5}+5a^{3}z^{3}+7a^{3}z+3a^{3}z^{-1}-az^{7}-7az^{5}+z^{5}a^{-1}-17az^{3}+5z^{3}a^{-1}-17az+7za^{-1}-5az^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-a^{4}z^{8}-2a^{2}z^{8}-z^{8}-a^{5}z^{7}+6a^{3}z^{7}+7az^{7}+5a^{4}z^{6}+11a^{2}z^{6}+6z^{6}+6a^{5}z^{5}-13a^{3}z^{5}-21az^{5}-2z^{5}a^{-1}-6a^{4}z^{4}-20a^{2}z^{4}-z^{4}a^{-2}-15z^{4}-10a^{5}z^{3}+16a^{3}z^{3}+33az^{3}+6z^{3}a^{-1}-z^{3}a^{-3}+a^{4}z^{2}+17a^{2}z^{2}-z^{2}a^{-4}+17z^{2}+4a^{5}z-11a^{3}z-22az-7za^{-1}-5a^{2}+a^{-4}-5+3a^{3}z^{-1}+5az^{-1}+2a^{-1}z^{-1}$ (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 χ 8                   1 -1 6                     0 4               2 1   -1 2             1       1 0           2 3       1 -2         1           1 -4       1 2           1 -6     1 1             0 -8     1               1 -10 1 1                 0 -12 1                   1

Integral Khovanov Homology (db, data source)

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

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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