L11n115

L11n114

L11n116

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Link Presentations

[edit Notes on L11n115's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X20,8,21,7 X22,17,5,18 X18,21,19,22 X13,17,14,16 X9,15,10,14 X15,11,16,10 X8,20,9,19 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, -7, 8, 11, -2, -6, 7, -8, 6, 4, -5, 9, -3, 5, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{5}+t(1)t(2)^{4}-4t(2)^{4}-2t(1)t(2)^{3}+3t(2)^{3}+3t(1)t(2)^{2}-2t(2)^{2}-4t(1)t(2)+t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{7/2}-3q^{5/2}+5q^{3/2}-7{\sqrt {q}}+{\frac {7}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{5}+2a^{5}z^{-1}-3z^{3}a^{3}-8za^{3}-5a^{3}z^{-1}+2z^{5}a+8z^{3}a+11za+6az^{-1}-3z^{3}a^{-1}-7za^{-1}-4a^{-1}z^{-1}+za^{-3}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$a^{5}z^{7}-5a^{5}z^{5}+9a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+2a^{4}z^{8}-8a^{4}z^{6}+8a^{4}z^{4}-a^{4}z^{2}+z^{2}a^{-4}-a^{4}-a^{-4}+a^{3}z^{9}+2a^{3}z^{7}-23a^{3}z^{5}+36a^{3}z^{3}+3z^{3}a^{-3}-22a^{3}z-3za^{-3}+5a^{3}z^{-1}+a^{-3}z^{-1}+6a^{2}z^{8}-22a^{2}z^{6}+2z^{6}a^{-2}+18a^{2}z^{4}-5z^{4}a^{-2}-2a^{2}z^{2}+8z^{2}a^{-2}-a^{2}-3a^{-2}+az^{9}+6az^{7}+5z^{7}a^{-1}-38az^{5}-20z^{5}a^{-1}+54az^{3}+30z^{3}a^{-1}-30az-18za^{-1}+6az^{-1}+4a^{-1}z^{-1}+4z^{8}-12z^{6}+5z^{4}+6z^{2}-3$ (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                 2   2 4               3 1   -2 2             4 2     2 0           4 4       0 -2         3 4 1       0 -4       3 4           1 -6     2 3             -1 -8   1 4               3 -10   1                 -1 -12 1                   1

Integral Khovanov Homology (db, data source)

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

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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