L11a71

L11a70

L11a72

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Link Presentations

[edit Notes on L11a71's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X16,8,17,7 X18,10,19,9 X8,18,9,17 X22,20,5,19 X20,14,21,13 X14,22,15,21 X10,16,11,15 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -5, 4, -9, 11, -2, 7, -8, 9, -3, 5, -4, 6, -7, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{5}-4uv^{4}+7uv^{3}-8uv^{2}+6uv-3u-3v^{5}+6v^{4}-8v^{3}+7v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$17q^{9/2}-19q^{7/2}+18q^{5/2}-{\frac {1}{q^{5/2}}}-15q^{3/2}+{\frac {2}{q^{3/2}}}+q^{17/2}-4q^{15/2}+9q^{13/2}-13q^{11/2}+10{\sqrt {q}}-{\frac {7}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+za^{-7}-2z^{5}a^{-5}-4z^{3}a^{-5}+a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+2z^{3}a^{-3}-za^{-3}-a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+3az-5za^{-1}+2az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}+4z^{5}a^{-9}-z^{3}a^{-9}+9z^{6}a^{-8}-9z^{4}a^{-8}+4z^{2}a^{-8}-a^{-8}+12z^{7}a^{-7}-16z^{5}a^{-7}+8z^{3}a^{-7}-2za^{-7}+10z^{8}a^{-6}-9z^{6}a^{-6}-5z^{4}a^{-6}+3z^{2}a^{-6}+5z^{9}a^{-5}+7z^{7}a^{-5}-32z^{5}a^{-5}+20z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+z^{10}a^{-4}+16z^{8}a^{-4}-46z^{6}a^{-4}+38z^{4}a^{-4}-15z^{2}a^{-4}+3a^{-4}+8z^{9}a^{-3}-14z^{7}a^{-3}-3z^{5}a^{-3}+5z^{3}a^{-3}+3za^{-3}-a^{-3}z^{-1}+z^{10}a^{-2}+8z^{8}a^{-2}-34z^{6}a^{-2}+36z^{4}a^{-2}-10z^{2}a^{-2}+3z^{9}a^{-1}+az^{7}-8z^{7}a^{-1}-5az^{5}+4z^{5}a^{-1}+9az^{3}+3z^{3}a^{-1}-7az-4za^{-1}+2az^{-1}+2a^{-1}z^{-1}+2z^{8}-6z^{6}+3z^{4}+4z^{2}-3$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     3   3 14                   6 1   -5 12                 7 3     4 10               10 6       -4 8             9 7         2 6           9 10           1 4         6 9             -3 2       5 10               5 0     2 5                 -3 -2     5                   5 -4 1 2                     -1 -6 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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