L10n49

L10n48

L10n50

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Link Presentations

[edit Notes on L10n49's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X20,14,7,13 X12,5,13,6 X3,10,4,11 X4,15,5,16 X11,16,12,17 X14,20,15,19 X17,2,18,3
Gauss code	{1, 10, -6, -7, 5, -3}, {3, -1, 2, 6, -8, -5, 4, -9, 7, 8, -10, -2, 9, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{4}-2u^{2}v^{3}-uv^{2}-2v+1}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {2}{q^{5/2}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}-z^{5}a^{7}-5z^{3}a^{7}-6za^{7}-2a^{7}z^{-1}+z^{7}a^{5}+6z^{5}a^{5}+11z^{3}a^{5}+8za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-5z^{3}a^{3}-6za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{11}z+a^{10}z^{2}+2a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+a^{8}z^{6}-3a^{8}z^{4}+a^{8}z^{2}+2a^{7}z^{7}-10a^{7}z^{5}+15a^{7}z^{3}-10a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}-4a^{6}z^{6}+3a^{6}z^{4}-a^{6}z^{2}+a^{6}+3a^{5}z^{7}-16a^{5}z^{5}+24a^{5}z^{3}-12a^{5}z+2a^{5}z^{-1}+a^{4}z^{8}-5a^{4}z^{6}+6a^{4}z^{4}-a^{4}z^{2}+a^{3}z^{7}-6a^{3}z^{5}+11a^{3}z^{3}-7a^{3}z+a^{3}z^{-1}$ (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 χ 0                 1 1 -2                   0 -4             2 1   1 -6           1 1     0 -8         1 1       0 -10       1 1         0 -12     1 1           0 -14     1             1 -16 1 1               0 -18 1                 1

Integral Khovanov Homology (db, data source)

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

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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