L11n112

L11n111

L11n113

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Link Presentations

[edit Notes on L11n112's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,18,8,19 X19,22,20,5 X9,21,10,20 X21,9,22,8 X11,17,12,16 X17,15,18,14 X15,11,16,10 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 6, -5, 9, -7, -2, 11, 8, -9, 7, -8, 3, -4, 5, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2(u-1)(v-1)}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$-4q^{9/2}+3q^{7/2}-3q^{5/2}-{\frac {1}{q^{5/2}}}+2q^{3/2}+{\frac {1}{q^{3/2}}}+q^{15/2}-2q^{13/2}+3q^{11/2}-{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}+a^{-7}z^{-1}-2z^{3}a^{-5}-5za^{-5}-4a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+10za^{-3}+6a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+3az-9za^{-1}+2az^{-1}-5a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-2}-z^{8}a^{-4}-z^{8}a^{-6}-z^{8}-az^{7}-2z^{7}a^{-1}-3z^{7}a^{-3}-4z^{7}a^{-5}-2z^{7}a^{-7}+7z^{6}a^{-2}+3z^{6}a^{-4}+z^{6}a^{-6}-z^{6}a^{-8}+6z^{6}+6az^{5}+16z^{5}a^{-1}+20z^{5}a^{-3}+18z^{5}a^{-5}+8z^{5}a^{-7}-10z^{4}a^{-2}+2z^{4}a^{-4}+8z^{4}a^{-6}+4z^{4}a^{-8}-8z^{4}-10az^{3}-33z^{3}a^{-1}-39z^{3}a^{-3}-24z^{3}a^{-5}-8z^{3}a^{-7}-7z^{2}a^{-4}-9z^{2}a^{-6}-4z^{2}a^{-8}+2z^{2}+7az+22za^{-1}+27za^{-3}+16za^{-5}+4za^{-7}+a^{-2}+3a^{-4}+3a^{-6}+a^{-8}+1-2az^{-1}-5a^{-1}z^{-1}-6a^{-3}z^{-1}-4a^{-5}z^{-1}-a^{-7}z^{-1}$ (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 16                       1 -1 14                     1   1 12                   2 1   -1 10                 2 1     1 8             1 2 2       1 6             2 2         0 4         1 2 2           1 2       1 2 2             -1 0       1 3               2 -2   1 1                   0 -4                         0 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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