L11n70

L11n69

L11n71

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Link Presentations

[edit Notes on L11n70's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,8,17,7 X17,22,18,5 X11,18,12,19 X13,20,14,21 X19,12,20,13 X21,14,22,15 X8,16,9,15 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -5, 7, -6, 8, 9, -3, -4, 5, -7, 6, -8, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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