L11n15

L11n14

L11n16

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Link Presentations

[edit Notes on L11n15's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,10,6,11 X8493 X11,22,12,5 X13,20,14,21 X19,14,20,15 X21,12,22,13 X9,18,10,19 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -10, 4, -6, 9, -7, 8, 11, -2, 3, 10, -8, 7, -9, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{19/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{13/2}}}+{\frac {1}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{9}-a^{9}z^{-1}+z^{3}a^{7}+3za^{7}+2a^{7}z^{-1}-za^{5}-a^{5}z^{-1}+a^{3}z^{-1}-za-az^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{10}+7z^{6}a^{10}-15z^{4}a^{10}+11z^{2}a^{10}-3a^{10}-z^{9}a^{9}+7z^{7}a^{9}-15z^{5}a^{9}+11z^{3}a^{9}-3za^{9}+a^{9}z^{-1}-2z^{8}a^{8}+14z^{6}a^{8}-31z^{4}a^{8}+26z^{2}a^{8}-7a^{8}-z^{9}a^{7}+7z^{7}a^{7}-16z^{5}a^{7}+16z^{3}a^{7}-8za^{7}+2a^{7}z^{-1}-z^{8}a^{6}+7z^{6}a^{6}-16z^{4}a^{6}+15z^{2}a^{6}-4a^{6}-z^{5}a^{5}+5z^{3}a^{5}-5za^{5}+a^{5}z^{-1}-za^{3}+a^{3}z^{-1}-a^{2}-za+az^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ 0                   1 1 -2                 1 2 1 -4               1 1 1 1 -6             1 2     1 -8           1 1 1     1 -10         1 2 1       0 -12       1 1           0 -14       1 1           0 -16   1 1               0 -18                     0 -20 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $i=0$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$

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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