L10n95

L10n94

L10n96

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Link Presentations

[edit Notes on L10n95's Link Presentations]

Planar diagram presentation	X8192 X18,12,19,11 X3,10,4,11 X19,2,20,3 X16,8,17,7 X9,13,10,20 X12,18,7,17 X4,13,5,14 X14,5,15,6 X6,15,1,16
Gauss code	{1, 4, -3, -8, 9, -10}, {5, -1, -6, 3, 2, -7}, {8, -9, 10, -5, 7, -2, -4, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(2)t(3)^{3}+t(1)^{2}t(3)^{2}-t(1)t(2)^{2}t(3)^{2}-t(1)t(3)^{2}-t(1)^{2}t(2)t(3)^{2}+2t(1)t(2)t(3)^{2}-t(2)t(3)^{2}+t(1)t(2)^{2}t(3)-t(2)^{2}t(3)+t(1)t(3)+t(1)^{2}t(2)t(3)-2t(1)t(2)t(3)+t(2)t(3)+t(1)t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-7}-2q^{-6}+4q^{-5}-5q^{-4}+6q^{-3}-5q^{-2}+2q+5q^{-1}-2$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{2}+a^{6}-a^{4}z^{4}-a^{4}z^{2}+a^{4}z^{-2}+a^{4}-2a^{2}z^{4}-6a^{2}z^{2}-2a^{2}z^{-2}-6a^{2}+2z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$z^{4}a^{8}-2z^{2}a^{8}+2z^{5}a^{7}-3z^{3}a^{7}+3z^{6}a^{6}-7z^{4}a^{6}+7z^{2}a^{6}-2a^{6}+2z^{7}a^{5}-3z^{5}a^{5}+3z^{3}a^{5}+z^{8}a^{4}-z^{6}a^{4}+3z^{4}a^{4}-3z^{2}a^{4}-a^{4}z^{-2}+3a^{4}+3z^{7}a^{3}-8z^{5}a^{3}+13z^{3}a^{3}-9za^{3}+2a^{3}z^{-1}+z^{8}a^{2}-4z^{6}a^{2}+14z^{4}a^{2}-21z^{2}a^{2}-2a^{2}z^{-2}+11a^{2}+z^{7}a-3z^{5}a+7z^{3}a-9za+2az^{-1}+3z^{4}-9z^{2}-z^{-2}+7$ (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 3                 2 2 1               1 1 0 -1             4 1   3 -3           3 3     0 -5         3 2       1 -7       2 3         1 -9     2 3           -1 -11     2             2 -13 1 2               -1 -15 1                 1

Integral Khovanov Homology (db, data source)

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

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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