L11n163

L11n162

L11n164

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Link Presentations

[edit Notes on L11n163's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X17,13,18,12 X14,5,15,6 X4,13,5,14 X11,19,12,18 X22,19,7,20 X20,15,21,16 X16,21,17,22 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, -6, 3, 5, -4, 8, -9, -3, 6, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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