L10n15

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Link Presentations

[edit Notes on L10n15's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X14,10,15,9 X8493 X5,13,6,12 X13,5,14,20 X16,12,17,11 X10,16,11,15 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, 4, -9, 8, 6, -7, -4, 9, -8, 10, -2, 3, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{4}-v^{3}+v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$6q^{9/2}-7q^{7/2}+6q^{5/2}-7q^{3/2}+{\frac {1}{q^{3/2}}}+2q^{13/2}-4q^{11/2}+4{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$za^{-7}+a^{-7}z^{-1}-z^{5}a^{-5}-4z^{3}a^{-5}-5za^{-5}-2a^{-5}z^{-1}+z^{7}a^{-3}+5z^{5}a^{-3}+8z^{3}a^{-3}+5za^{-3}-z^{5}a^{-1}-3z^{3}a^{-1}-za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3z^{2}a^{-8}-2a^{-8}+z^{5}a^{-7}+3z^{3}a^{-7}-2za^{-7}+a^{-7}z^{-1}+4z^{6}a^{-6}-9z^{4}a^{-6}+12z^{2}a^{-6}-5a^{-6}+5z^{7}a^{-5}-15z^{5}a^{-5}+18z^{3}a^{-5}-9za^{-5}+2a^{-5}z^{-1}+2z^{8}a^{-4}-12z^{4}a^{-4}+12z^{2}a^{-4}-3a^{-4}+8z^{7}a^{-3}-27z^{5}a^{-3}+26z^{3}a^{-3}-10za^{-3}+2z^{8}a^{-2}-3z^{6}a^{-2}-6z^{4}a^{-2}+5z^{2}a^{-2}+a^{-2}+3z^{7}a^{-1}-11z^{5}a^{-1}+11z^{3}a^{-1}-3za^{-1}-a^{-1}z^{-1}+z^{6}-3z^{4}+2z^{2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 χ 14                 2 -2 12               2   2 10             4 2   -2 8           3 2     1 6         3 4       1 4       4 3         1 2     2 5           3 0   1 2             -1 -2   2               2 -4 1                 -1

Integral Khovanov Homology (db, data source)

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

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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