L10n94

L10n93

L10n95

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Link Presentations

[edit Notes on L10n94's Link Presentations]

Planar diagram presentation	X8192 X18,10,19,9 X6,18,1,17 X7,17,8,16 X3,10,4,11 X14,6,15,5 X4,14,5,13 X11,13,12,20 X15,7,16,12 X19,3,20,2
Gauss code	{1, 10, -5, -7, 6, -3}, {-4, -1, 2, 5, -8, 9}, {7, -6, -9, 4, 3, -2, -10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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