L10n97

L10n96

L10n98

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Link Presentations

[edit Notes on L10n97's Link Presentations]

Planar diagram presentation	X6172 X2536 X20,13,15,14 X3,11,4,10 X9,1,10,4 X7,17,8,16 X15,5,16,8 X18,11,19,12 X12,19,13,20 X14,17,9,18
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 8, -9, 3, -10}, {-7, 6, 10, -8, 9, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(4)^{2}t(3)^{2}+t(1)t(4)t(3)^{2}+t(2)t(4)^{2}t(3)+t(1)t(3)-t(1)t(4)t(3)-t(2)t(4)t(3)-t(1)t(2)+t(2)t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$-{\frac {2}{q^{9/2}}}+{\frac {1}{q^{7/2}}}-q^{5/2}-{\frac {3}{q^{5/2}}}-{\frac {1}{q^{13/2}}}+{\frac {1}{q^{11/2}}}-{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z^{-3}+3a^{5}z+3a^{5}z^{-1}-a^{3}z^{5}-6a^{3}z^{3}-3a^{3}z^{-3}-13a^{3}z-10a^{3}z^{-1}+az^{5}+7az^{3}+3az^{-3}-z^{3}a^{-1}-a^{-1}z^{-3}+14az+11az^{-1}-4za^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{7}+4z^{3}a^{7}-3za^{7}+a^{7}z^{-1}-z^{6}a^{6}+3z^{4}a^{6}-a^{6}-z^{7}a^{5}+4z^{5}a^{5}-4z^{3}a^{5}+3za^{5}-3a^{5}z^{-1}+a^{5}z^{-3}-2z^{6}a^{4}+10z^{4}a^{4}-16z^{2}a^{4}-3a^{4}z^{-2}+11a^{4}-z^{7}a^{3}+7z^{5}a^{3}-18z^{3}a^{3}+21za^{3}-12a^{3}z^{-1}+3a^{3}z^{-3}-2z^{6}a^{2}+15z^{4}a^{2}-33z^{2}a^{2}-6a^{2}z^{-2}+24a^{2}-z^{7}a+9z^{5}a-25z^{3}a+28za-14az^{-1}+3az^{-3}-z^{6}+8z^{4}-17z^{2}-3z^{-2}+13-z^{7}a^{-1}+7z^{5}a^{-1}-15z^{3}a^{-1}+13za^{-1}-6a^{-1}z^{-1}+a^{-1}z^{-3}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                     1 1 4                     1 1 2                 1     1 0             3         3 -2           1 4 1       2 -4         3             3 -6       1 4 1           2 -8     1 1 1             1 -10     1                 1 -12 1 1                   0 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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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