L10n54

L10n53

L10n55

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Link Presentations

[edit Notes on L10n54's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X20,12,9,11 X2,9,3,10 X17,5,18,4 X5,19,6,18 X6,14,7,13 X14,8,15,7 X8,16,1,15 X19,17,20,16
Gauss code	{1, -4, 2, 5, -6, -7, 8, -9}, {4, -1, 3, -2, 7, -8, 9, 10, -5, 6, -10, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{3}t(1)^{3}-t(2)^{2}t(1)^{3}-t(2)^{3}t(1)^{2}-t(1)-t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-2q^{9/2}+2q^{7/2}-2q^{5/2}+q^{3/2}+q^{15/2}-2q^{13/2}+q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-za^{-9}+z^{5}a^{-7}+5z^{3}a^{-7}+5za^{-7}-z^{7}a^{-5}-6z^{5}a^{-5}-11z^{3}a^{-5}-8za^{-5}-a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+6za^{-3}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-4z^{4}a^{-8}+2z^{2}a^{-8}+2z^{7}a^{-7}-10z^{5}a^{-7}+12z^{3}a^{-7}-3za^{-7}+z^{8}a^{-6}-4z^{6}a^{-6}+2z^{4}a^{-6}+z^{2}a^{-6}+3z^{7}a^{-5}-16z^{5}a^{-5}+23z^{3}a^{-5}-10za^{-5}+a^{-5}z^{-1}+z^{8}a^{-4}-5z^{6}a^{-4}+6z^{4}a^{-4}-z^{2}a^{-4}-a^{-4}+z^{7}a^{-3}-6z^{5}a^{-3}+11z^{3}a^{-3}-7za^{-3}+a^{-3}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 χ 16               1 -1 14             1   1 12           1 2   1 10         1 1 1   1 8       1 1       0 6     1 1         0 4   1 2           1 2                 0 0 1               1

Integral Khovanov Homology (db, data source)

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

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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