L11n378

L11n377

L11n379

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Link Presentations

[edit Notes on L11n378's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X15,1,16,4 X5,10,6,11 X3849 X22,18,19,17 X11,20,12,21 X19,12,20,13 X18,22,5,21 X9,16,10,17 X2,14,3,13
Gauss code	{1, -11, -5, 3}, {-8, 7, 9, -6}, {-4, -1, 2, 5, -10, 4, -7, 8, 11, -2, -3, 10, 6, -9}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}}}$ (db)
Jones polynomial	$-q^{-7}+q^{-6}-2q^{-5}+3q^{-4}-2q^{-3}+3q^{-2}-q^{-1}+3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}z^{-2}-2a^{6}+z^{4}a^{4}+4z^{2}a^{4}+4a^{4}z^{-2}+7a^{4}-3z^{2}a^{2}-5a^{2}z^{-2}-8a^{2}+2z^{-2}+3$ (db)
Kauffman polynomial	$a^{7}z^{7}-6a^{7}z^{5}+11a^{7}z^{3}-6a^{7}z+a^{7}z^{-1}+a^{6}z^{8}-5a^{6}z^{6}+7a^{6}z^{4}-4a^{6}z^{2}-a^{6}z^{-2}+3a^{6}+3a^{5}z^{7}-16a^{5}z^{5}+27a^{5}z^{3}-19a^{5}z+5a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{6}+6a^{4}z^{4}-11a^{4}z^{2}-4a^{4}z^{-2}+10a^{4}+2a^{3}z^{7}-10a^{3}z^{5}+19a^{3}z^{3}-21a^{3}z+9a^{3}z^{-1}+a^{2}z^{6}-a^{2}z^{4}-7a^{2}z^{2}-5a^{2}z^{-2}+11a^{2}+3az^{3}-8az+5az^{-1}-2z^{-2}+5$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 χ 1               3 3 -1             2 4 2 -3           1   1 2 -5         1 2     1 -7       2 1       1 -9       1         1 -11   1 2           -1 -13                 0 -15 1               -1

Integral Khovanov Homology (db, data source)

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

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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