L11n65

L11n64

L11n66

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Link Presentations

[edit Notes on L11n65's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X17,20,18,21 X11,18,12,19 X19,12,20,13 X15,22,16,5 X21,16,22,17 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)t(2)^{3}+t(2)^{3}-2t(1)t(2)^{2}-2t(2)+t(1)+2}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{25/2}}}-{\frac {1}{q^{23/2}}}+{\frac {1}{q^{21/2}}}-{\frac {2}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {3}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+2a^{11}z+2a^{11}z^{-1}-a^{9}z^{-1}-a^{7}z^{5}-3a^{7}z^{3}+a^{7}z^{-1}-a^{5}z^{5}-4a^{5}z^{3}-4a^{5}z-a^{5}z^{-1}$ (db)
Kauffman polynomial	$a^{14}z^{8}-7a^{14}z^{6}+15a^{14}z^{4}-12a^{14}z^{2}+3a^{14}+a^{13}z^{9}-7a^{13}z^{7}+14a^{13}z^{5}-10a^{13}z^{3}+4a^{13}z-a^{13}z^{-1}+2a^{12}z^{8}-16a^{12}z^{6}+37a^{12}z^{4}-29a^{12}z^{2}+7a^{12}+a^{11}z^{9}-8a^{11}z^{7}+19a^{11}z^{5}-17a^{11}z^{3}+9a^{11}z-2a^{11}z^{-1}+a^{10}z^{8}-8a^{10}z^{6}+20a^{10}z^{4}-16a^{10}z^{2}+4a^{10}+2a^{9}z^{5}-4a^{9}z^{3}+3a^{9}z-a^{9}z^{-1}+2a^{8}z^{6}-4a^{8}z^{4}+a^{7}z^{7}-2a^{7}z^{5}-a^{7}z^{3}+2a^{7}z-a^{7}z^{-1}+a^{6}z^{6}-2a^{6}z^{4}-a^{6}z^{2}+a^{6}+a^{5}z^{5}-4a^{5}z^{3}+4a^{5}z-a^{5}z^{-1}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     1 1 0 -8                   2     2 -10                 1 1     0 -12             1 3 2       0 -14             1 2         1 -16         1 3 2           0 -18       1   1             2 -20       1 2               -1 -22   1 1                   0 -24                         0 -26 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $i=-2$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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