L10n92

L10n91

L10n93

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Link Presentations

[edit Notes on L10n92's Link Presentations]

Planar diagram presentation	X8192 X18,10,19,9 X17,6,18,1 X7,17,8,16 X3,10,4,11 X5,14,6,15 X13,4,14,5 X11,13,12,20 X15,7,16,12 X2,19,3,20
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 $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{2}t(3)^{3}-t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}+t(1)t(2)t(3)^{2}-t(1)^{2}t(3)+t(1)t(3)-t(1)t(2)t(3)-t(1)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{-5}+2q^{-4}-2q^{-3}+q^{2}+3q^{-2}-q-2q^{-1}+3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+4a^{4}z^{2}+a^{4}z^{-2}+4a^{4}-a^{2}z^{6}-6a^{2}z^{4}-12a^{2}z^{2}-2a^{2}z^{-2}-9a^{2}+z^{4}+4z^{2}+z^{-2}+5$ (db)
Kauffman polynomial	$a^{6}z^{6}-5a^{6}z^{4}+6a^{6}z^{2}-a^{6}+a^{5}z^{7}-4a^{5}z^{5}+3a^{5}z^{3}+a^{4}z^{8}-5a^{4}z^{6}+9a^{4}z^{4}-10a^{4}z^{2}-a^{4}z^{-2}+5a^{4}+2a^{3}z^{7}-9a^{3}z^{5}+13a^{3}z^{3}-9a^{3}z+2a^{3}z^{-1}+a^{2}z^{8}-6a^{2}z^{6}+16a^{2}z^{4}-21a^{2}z^{2}+z^{2}a^{-2}-2a^{2}z^{-2}+11a^{2}-a^{-2}+az^{7}-5az^{5}+11az^{3}+z^{3}a^{-1}-9az+2az^{-1}+2z^{4}-4z^{2}-z^{-2}+5$ (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 5                 1 1 3               1 1 0 1             2     2 -1           2 3     1 -3         1         1 -5       1 2         1 -7     1 1           0 -9     1             1 -11 1 1               0 -13 1                 1

Integral Khovanov Homology (db, data source)

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

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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