L11n153

L11n152

L11n154

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Link Presentations

[edit Notes on L11n153's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X7,14,8,15 X13,20,14,21 X19,1,20,6 X18,11,19,12 X5,12,6,13 X15,22,16,7 X4,18,5,17 X21,16,22,17
Gauss code	{1, -2, 3, -10, -8, 6}, {-4, -1, 2, -3, 7, 8, -5, 4, -9, 11, 10, -7, -6, 5, -11, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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