L11n452

L11n451

L11n453

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Link Presentations

[edit Notes on L11n452's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,20,12,21 X13,19,14,22 X21,18,22,9 X17,13,18,12 X8,16,5,15 X14,8,15,7 X19,17,20,16 X2536 X4,9,1,10
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 $u$, $v$, $w$, ...)	${\frac {(t(3)-1)(t(4)-1)^{2}(t(2)t(4)-t(1))}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}t(4)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}+{\frac {2}{q^{9/2}}}-q^{7/2}-{\frac {6}{q^{7/2}}}-q^{5/2}+{\frac {5}{q^{5/2}}}+2q^{3/2}-{\frac {7}{q^{3/2}}}-{\frac {1}{q^{11/2}}}-6{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-3}+a^{5}z+2a^{5}z^{-1}-3a^{3}z^{3}-3a^{3}z^{-3}+z^{3}a^{-3}-8a^{3}z-8a^{3}z^{-1}+3za^{-3}+a^{-3}z^{-1}+2az^{5}-z^{5}a^{-1}+9az^{3}+3az^{-3}-7z^{3}a^{-1}-a^{-1}z^{-3}+15az+11az^{-1}-11za^{-1}-6a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{5}z^{7}-5a^{5}z^{5}+10a^{5}z^{3}-a^{5}z^{-3}-10a^{5}z+5a^{5}z^{-1}+2a^{4}z^{8}-7a^{4}z^{6}+z^{6}a^{-4}+3a^{4}z^{4}-5z^{4}a^{-4}+7a^{4}z^{2}+4z^{2}a^{-4}+3a^{4}z^{-2}-9a^{4}-a^{-4}+a^{3}z^{9}+3a^{3}z^{7}+z^{7}a^{-3}-29a^{3}z^{5}-7z^{5}a^{-3}+45a^{3}z^{3}+11z^{3}a^{-3}-3a^{3}z^{-3}-34a^{3}z-9za^{-3}+14a^{3}z^{-1}+2a^{-3}z^{-1}+6a^{2}z^{8}-23a^{2}z^{6}+12a^{2}z^{4}-7z^{4}a^{-2}+19a^{2}z^{2}+15z^{2}a^{-2}+6a^{2}z^{-2}-21a^{2}-6a^{-2}+az^{9}+5az^{7}+4z^{7}a^{-1}-44az^{5}-27z^{5}a^{-1}+75az^{3}+51z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-50az-35za^{-1}+18az^{-1}+11a^{-1}z^{-1}+4z^{8}-17z^{6}+7z^{4}+23z^{2}+3z^{-2}-18$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                         0 6                 2 1 1   2 4               2 1       -1 2             5 2 1       4 0           5 8 1         2 -2         2 3 4           3 -4       3 5               2 -6     3 2                 1 -8   1 5                   4 -10   1                     -1 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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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