L11n183

L11n182

L11n184

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Link Presentations

[edit Notes on L11n183's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X21,1,22,6 X7,18,8,19 X3,10,4,11 X15,12,16,13 X5,14,6,15 X13,4,14,5 X11,16,12,17 X17,22,18,7 X2,20,3,19
Gauss code	{1, -11, -5, 8, -7, 3}, {-4, -1, 2, 5, -9, 6, -8, 7, -6, 9, -10, 4, 11, -2, -3, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(2)^{2}t(1)^{2}-3t(2)t(1)^{2}+t(1)^{2}-4t(2)^{2}t(1)+7t(2)t(1)-4t(1)+t(2)^{2}-3t(2)+2}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {2}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {9}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {7}{q^{15/2}}}+{\frac {5}{q^{17/2}}}-{\frac {3}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)-a^{9}z+a^{7}z^{5}+2a^{7}z^{3}+a^{7}z-a^{7}z^{-1}+a^{5}z^{5}+2a^{5}z^{3}+3a^{5}z+3a^{5}z^{-1}-2a^{3}z^{3}-4a^{3}z-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+3z^{4}a^{12}-2z^{2}a^{12}-3z^{7}a^{11}+10z^{5}a^{11}-8z^{3}a^{11}+za^{11}-3z^{8}a^{10}+8z^{6}a^{10}-4z^{4}a^{10}+z^{2}a^{10}-z^{9}a^{9}-3z^{7}a^{9}+14z^{5}a^{9}-9z^{3}a^{9}+za^{9}-5z^{8}a^{8}+12z^{6}a^{8}-8z^{4}a^{8}+2z^{2}a^{8}+a^{8}-z^{9}a^{7}-2z^{7}a^{7}+7z^{5}a^{7}-6z^{3}a^{7}+2za^{7}-a^{7}z^{-1}-2z^{8}a^{6}+2z^{6}a^{6}-z^{4}a^{6}-4z^{2}a^{6}+3a^{6}-2z^{7}a^{5}+3z^{5}a^{5}-8z^{3}a^{5}+7za^{5}-3a^{5}z^{-1}-z^{6}a^{4}-3z^{2}a^{4}+3a^{4}-3z^{3}a^{3}+5za^{3}-2a^{3}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   2 2 -4                 2 1 -1 -6               5 1   4 -8             4 3     -1 -10           5 4       1 -12         4 4         0 -14       3 5           -2 -16     2 4             2 -18   1 3               -2 -20   2                 2 -22 1                   -1

Integral Khovanov Homology (db, data source)

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

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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