L10n81

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Link Presentations

[edit Notes on L10n81's Link Presentations]

Planar diagram presentation	X6172 X2,16,3,15 X3,10,4,11 X5,14,6,15 X11,20,12,13 X13,12,14,5 X19,1,20,4 X8,17,9,18 X16,7,17,8 X18,9,19,10
Gauss code	{1, -2, -3, 7}, {-4, -1, 9, -8, 10, 3, -5, 6}, {-6, 4, 2, -9, 8, -10, -7, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(3)^{2}t(2)^{3}-t(1)t(3)^{3}t(2)^{2}+2t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}-t(1)t(3)t(2)^{2}+2t(3)t(2)^{2}-2t(1)t(3)^{2}t(2)+t(3)^{2}t(2)+2t(1)t(3)t(2)-2t(3)t(2)+t(2)-t(1)t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-10}-2q^{-9}+5q^{-8}-6q^{-7}+6q^{-6}-6q^{-5}+6q^{-4}-2q^{-3}+2q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{4}+3a^{8}z^{2}+a^{8}z^{-2}+4a^{8}-a^{6}z^{6}-5a^{6}z^{4}-11a^{6}z^{2}-2a^{6}z^{-2}-11a^{6}+2a^{4}z^{4}+7a^{4}z^{2}+a^{4}z^{-2}+7a^{4}$ (db)
Kauffman polynomial	$a^{12}z^{4}-2a^{12}z^{2}+a^{12}+2a^{11}z^{5}-2a^{11}z^{3}+3a^{10}z^{6}-4a^{10}z^{4}+2a^{10}z^{2}+2a^{9}z^{7}-2a^{9}z^{3}+a^{8}z^{8}+4a^{8}z^{4}-7a^{8}z^{2}-a^{8}z^{-2}+5a^{8}+3a^{7}z^{7}-5a^{7}z^{5}+9a^{7}z^{3}-11a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}-3a^{6}z^{6}+12a^{6}z^{4}-20a^{6}z^{2}-2a^{6}z^{-2}+13a^{6}+a^{5}z^{7}-3a^{5}z^{5}+9a^{5}z^{3}-11a^{5}z+2a^{5}z^{-1}+3a^{4}z^{4}-9a^{4}z^{2}-a^{4}z^{-2}+8a^{4}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                 2 2 -5               2 2 0 -7             4     4 -9           2 2     0 -11         4 4       0 -13       3 3         0 -15     2 3           -1 -17     3             3 -19 1 2               -1 -21 1                 1

Integral Khovanov Homology (db, data source)

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

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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