L11n260

L11n259

L11n261

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Link Presentations

[edit Notes on L11n260's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,12,19,11 X19,22,20,9 X15,20,16,21 X21,16,22,17 X12,18,13,17 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -9, -4, 3, -7, 8, 9, -5, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-t(2)t(3)^{3}+t(3)^{3}+2t(1)t(3)^{2}-2t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}-3t(3)^{2}-2t(1)t(3)+3t(1)t(2)t(3)-2t(2)t(3)+2t(3)+t(1)-t(1)t(2)+t(2)-1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$2q^{4}-4q^{3}+8q^{2}-8q+10-8q^{-1}+7q^{-2}-4q^{-3}+q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{-4}z^{-2}+a^{-4}+a^{2}z^{4}+z^{4}a^{-2}+a^{2}z^{2}-2a^{-2}z^{-2}-3a^{-2}-z^{6}-3z^{4}-2z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$a^{4}z^{4}+3z^{4}a^{-4}-5z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+z^{7}a^{-3}+4a^{3}z^{5}+z^{5}a^{-3}-3a^{3}z^{3}-3za^{-3}+2a^{-3}z^{-1}+z^{8}a^{-2}+7a^{2}z^{6}+3z^{6}a^{-2}-10a^{2}z^{4}-4z^{4}a^{-2}+3a^{2}z^{2}-3z^{2}a^{-2}-2a^{-2}z^{-2}+5a^{-2}+5az^{7}+6z^{7}a^{-1}-3az^{5}-6z^{5}a^{-1}-3az^{3}-3za^{-1}+2a^{-1}z^{-1}+z^{8}+10z^{6}-18z^{4}+5z^{2}-z^{-2}+3$ (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   \ -4 -3 -2 -1 0 1 2 3 4 χ 9                 2 2 7               3 1 -2 5             5 1   4 3           3 3     0 1         7 5       2 -1       4 6         2 -3     3 4           -1 -5   1 4             3 -7   3               -3 -9 1                 1

Integral Khovanov Homology (db, data source)

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

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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