L11n117

L11n116

L11n118

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Link Presentations

[edit Notes on L11n117's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X17,21,18,20 X13,19,14,18 X19,15,20,14 X4,21,1,22 X10,5,11,6 X12,3,13,4 X22,11,5,12 X2,9,3,10 X8,15,9,16
Gauss code	{1, -10, 8, -6}, {7, -1, 2, -11, 10, -7, 9, -8, -4, 5, 11, -2, -3, 4, -5, 3, 6, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{5}+2t(1)t(2)^{4}-4t(2)^{4}-3t(1)t(2)^{3}+5t(2)^{3}+5t(1)t(2)^{2}-3t(2)^{2}-4t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$2{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {10}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {5}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{7}+za^{7}-a^{7}z^{-1}-z^{5}a^{5}-z^{3}a^{5}+3za^{5}+4a^{5}z^{-1}-2z^{5}a^{3}-7z^{3}a^{3}-9za^{3}-4a^{3}z^{-1}+2z^{3}a+3za+az^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{9}+4z^{5}a^{9}-5z^{3}a^{9}+2za^{9}-3z^{8}a^{8}+13z^{6}a^{8}-17z^{4}a^{8}+6z^{2}a^{8}+a^{8}-2z^{9}a^{7}+3z^{7}a^{7}+10z^{5}a^{7}-16z^{3}a^{7}+5za^{7}-a^{7}z^{-1}-9z^{8}a^{6}+32z^{6}a^{6}-28z^{4}a^{6}+3z^{2}a^{6}+4a^{6}-2z^{9}a^{5}-4z^{7}a^{5}+30z^{5}a^{5}-31z^{3}a^{5}+13za^{5}-4a^{5}z^{-1}-6z^{8}a^{4}+14z^{6}a^{4}-2z^{4}a^{4}-10z^{2}a^{4}+7a^{4}-8z^{7}a^{3}+23z^{5}a^{3}-25z^{3}a^{3}+14za^{3}-4a^{3}z^{-1}-5z^{6}a^{2}+9z^{4}a^{2}-10z^{2}a^{2}+4a^{2}-z^{5}a-5z^{3}a+4za-az^{-1}-3z^{2}+1$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 2                   2 -2 0                 3   3 -2               5 3   -2 -4             5 2     3 -6           4 5       1 -8         6 5         1 -10       3 5           2 -12     2 5             -3 -14   1 3               2 -16   2                 -2 -18 1                   1

Integral Khovanov Homology (db, data source)

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

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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