L11n10

L11n9

L11n11

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Link Presentations

[edit Notes on L11n10's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,10,6,11 X3849 X13,22,14,5 X11,20,12,21 X21,12,22,13 X19,14,20,15 X9,18,10,19 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -10, 4, -7, 8, -6, 9, -11, -2, 3, 10, -9, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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