L10n111

L10n110

L10n112

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Link Presentations

[edit Notes on L10n111's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,17,14,20 X19,11,20,16 X7,19,8,18 X15,8,16,9 X9,14,10,15 X17,5,18,10 X2536 X4,11,1,12
Gauss code	{1, -9, 2, -10}, {-8, 5, -4, 3}, {9, -1, -5, 6, -7, 8}, {10, -2, -3, 7, -6, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(3)t(4)-1)(t(3)t(1)+t(4)t(1)-t(1)-t(2)t(3)-t(2)t(4)+t(2)t(3)t(4))}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$-q^{7/2}+q^{5/2}-q^{3/2}-{\frac {2}{\sqrt {q}}}-{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}-{\frac {2}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$za^{5}+2a^{5}z^{-1}+a^{5}z^{-3}-2z^{3}a^{3}-8za^{3}-7a^{3}z^{-1}-3a^{3}z^{-3}+z^{5}a+6z^{3}a+10za+8az^{-1}+3az^{-3}-2za^{-1}-3a^{-1}z^{-1}-a^{-1}z^{-3}-za^{-3}$ (db)
Kauffman polynomial	$a^{5}z^{7}-6a^{5}z^{5}+11a^{5}z^{3}-a^{5}z^{-3}-10a^{5}z+5a^{5}z^{-1}+a^{4}z^{8}-5a^{4}z^{6}+3a^{4}z^{4}+8a^{4}z^{2}+3a^{4}z^{-2}-10a^{4}+3a^{3}z^{7}-20a^{3}z^{5}+z^{5}a^{-3}+39a^{3}z^{3}-4z^{3}a^{-3}-3a^{3}z^{-3}-31a^{3}z+2za^{-3}+12a^{3}z^{-1}+a^{2}z^{8}-4a^{2}z^{6}+z^{6}a^{-2}-6a^{2}z^{4}-4z^{4}a^{-2}+26a^{2}z^{2}+2z^{2}a^{-2}+6a^{2}z^{-2}-19a^{2}+3az^{7}+z^{7}a^{-1}-21az^{5}-6z^{5}a^{-1}+43az^{3}+11z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-35az-12za^{-1}+12az^{-1}+5a^{-1}z^{-1}+2z^{6}-13z^{4}+20z^{2}+3z^{-2}-10$ (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 3 4 χ 8                     1 1 6                   1 1 0 4               1 1     0 2             1 1 1     1 0           1 4 1       2 -2         2 1 3         4 -4       1 5 1           3 -6     1 1 3             3 -8   1 2                 1 -10                       0 -12 1                     1

Integral Khovanov Homology (db, data source)

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

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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