L11n324

L11n323

L11n325

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Link Presentations

[edit Notes on L11n324's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X11,21,12,20 X19,22,20,13 X13,12,14,5 X17,1,18,4 X21,11,22,10
Gauss code	{1, -4, -3, 10}, {-2, -1, 5, 3, -6, 11, -7, 9}, {-9, 2, 4, -5, -10, 6, -8, 7, -11, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(3)^{2}t(2)^{3}+t(3)t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-2t(1)t(3)^{2}t(2)^{2}+2t(3)^{2}t(2)^{2}+t(1)t(3)t(2)^{2}-2t(3)t(2)^{2}+t(2)^{2}-t(1)t(3)^{3}t(2)+2t(1)t(3)^{2}t(2)-t(3)^{2}t(2)-2t(1)t(3)t(2)+2t(3)t(2)-t(2)-t(1)t(3)^{2}+t(1)t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-8}+2q^{-7}-3q^{-6}+6q^{-5}-7q^{-4}+8q^{-3}-6q^{-2}+2q+6q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{6}-3z^{2}a^{6}-2a^{6}+z^{6}a^{4}+5z^{4}a^{4}+10z^{2}a^{4}+a^{4}z^{-2}+7a^{4}-3z^{4}a^{2}-10z^{2}a^{2}-2a^{2}z^{-2}-9a^{2}+2z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$a^{9}z^{5}-3a^{9}z^{3}+a^{9}z+2a^{8}z^{6}-6a^{8}z^{4}+4a^{8}z^{2}-a^{8}+2a^{7}z^{7}-4a^{7}z^{5}+a^{7}z+2a^{6}z^{8}-5a^{6}z^{6}+4a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{9}-a^{5}z^{7}-2a^{5}z^{5}+8a^{5}z^{3}-3a^{5}z+4a^{4}z^{8}-16a^{4}z^{6}+32a^{4}z^{4}-26a^{4}z^{2}-a^{4}z^{-2}+9a^{4}+a^{3}z^{9}-2a^{3}z^{7}+2a^{3}z^{5}+6a^{3}z^{3}-8a^{3}z+2a^{3}z^{-1}+2a^{2}z^{8}-9a^{2}z^{6}+25a^{2}z^{4}-31a^{2}z^{2}-2a^{2}z^{-2}+13a^{2}+az^{7}-az^{5}+az^{3}-5az+2az^{-1}+3z^{4}-8z^{2}-z^{-2}+6$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   2 2 1                 1   -1 -1               5 2   3 -3             4 4     0 -5           4 2       2 -7         3 4         1 -9       3 4           -1 -11     1 4             3 -13   1 2               -1 -15   1                 1 -17 1                   -1

Integral Khovanov Homology (db, data source)

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

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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