L11a156

L11a155

L11a157

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Link Presentations

[edit Notes on L11a156's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X14,6,15,5 X18,14,19,13 X20,17,21,18 X16,21,17,22 X12,20,13,19 X2738 X4,12,5,11 X6,16,1,15
Gauss code	{1, -9, 2, -10, 4, -11}, {9, -1, 3, -2, 10, -8, 5, -4, 11, -7, 6, -5, 8, -6, 7, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{4}-5u^{2}v^{3}+6u^{2}v^{2}-2u^{2}v-uv^{4}+6uv^{3}-9uv^{2}+6uv-u-2v^{3}+6v^{2}-5v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+6q^{13/2}-11q^{11/2}+14q^{9/2}-16q^{7/2}+16q^{5/2}-14q^{3/2}+10{\sqrt {q}}-{\frac {7}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}-2z^{5}a^{-5}-6z^{3}a^{-5}-5za^{-5}-a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+7z^{3}a^{-3}+7za^{-3}+2a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+2az-5za^{-1}+az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-3z^{9}a^{-1}-7z^{9}a^{-3}-4z^{9}a^{-5}-8z^{8}a^{-2}-12z^{8}a^{-4}-7z^{8}a^{-6}-3z^{8}-az^{7}+5z^{7}a^{-1}+10z^{7}a^{-3}-3z^{7}a^{-5}-7z^{7}a^{-7}+35z^{6}a^{-2}+39z^{6}a^{-4}+10z^{6}a^{-6}-5z^{6}a^{-8}+11z^{6}+4az^{5}+11z^{5}a^{-1}+20z^{5}a^{-3}+25z^{5}a^{-5}+9z^{5}a^{-7}-3z^{5}a^{-9}-35z^{4}a^{-2}-34z^{4}a^{-4}-6z^{4}a^{-6}+4z^{4}a^{-8}-z^{4}a^{-10}-12z^{4}-6az^{3}-23z^{3}a^{-1}-34z^{3}a^{-3}-26z^{3}a^{-5}-6z^{3}a^{-7}+3z^{3}a^{-9}+11z^{2}a^{-2}+11z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}a^{-8}+z^{2}a^{-10}+4z^{2}+4az+12za^{-1}+15za^{-3}+9za^{-5}+za^{-7}-za^{-9}-a^{-2}-az^{-1}-2a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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 5 6 7 χ 18                       1 -1 16                     2   2 14                   4 1   -3 12                 7 2     5 10               7 4       -3 8             9 7         2 6           8 8           0 4         6 8             -2 2       5 9               4 0     2 5                 -3 -2   1 5                   4 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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