L11n228

L11n227

L11n229

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Link Presentations

[edit Notes on L11n228's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X17,9,18,22 X9,21,10,20 X6,13,7,14 X14,7,15,8 X8,15,1,16 X19,5,20,4 X5,19,6,18 X21,17,22,16
Gauss code	{1, -2, 3, 9, -10, -6, 7, -8}, {-5, -1, 2, -3, 6, -7, 8, 11, -4, 10, -9, 5, -11, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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