L11n60

L11n59

L11n61

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Link Presentations

[edit Notes on L11n60's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X11,19,12,18 X22,19,5,20 X20,15,21,16 X16,21,17,22 X17,13,18,12 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -4, 8, 9, -3, 6, -7, -8, 4, 5, -6, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{5}+t(1)t(2)^{4}-3t(2)^{4}-4t(1)t(2)^{3}+6t(2)^{3}+6t(1)t(2)^{2}-4t(2)^{2}-3t(1)t(2)+t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{3/2}+4{\sqrt {q}}-{\frac {7}{\sqrt {q}}}+{\frac {9}{q^{3/2}}}-{\frac {11}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {6}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+6za^{5}+4a^{5}z^{-1}-2z^{5}a^{3}-7z^{3}a^{3}-10za^{3}-4a^{3}z^{-1}+3z^{3}a+4za+az^{-1}-za^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+3a^{8}z^{2}-a^{8}+3a^{7}z^{7}-9a^{7}z^{5}+8a^{7}z^{3}-4a^{7}z+a^{7}z^{-1}+3a^{6}z^{8}-4a^{6}z^{6}-9a^{6}z^{4}+11a^{6}z^{2}-4a^{6}+a^{5}z^{9}+7a^{5}z^{7}-29a^{5}z^{5}+29a^{5}z^{3}-15a^{5}z+4a^{5}z^{-1}+6a^{4}z^{8}-11a^{4}z^{6}-4a^{4}z^{4}+14a^{4}z^{2}-7a^{4}+a^{3}z^{9}+6a^{3}z^{7}-23a^{3}z^{5}+28a^{3}z^{3}-14a^{3}z+4a^{3}z^{-1}+3a^{2}z^{8}-6a^{2}z^{6}+6a^{2}z^{4}+4a^{2}z^{2}-4a^{2}+2az^{7}-3az^{5}+8az^{3}+z^{3}a^{-1}-4az-za^{-1}+az^{-1}+4z^{4}-2z^{2}-1$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 4                   1 1 2                 3   -3 0               4 1   3 -2             6 4     -2 -4           5 3       2 -6         4 6         2 -8       5 5           0 -10     2 5             3 -12   1 4               -3 -14   2                 2 -16 1                   -1

Integral Khovanov Homology (db, data source)

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

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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