L11n213

L11n212

L11n214

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Link Presentations

[edit Notes on L11n213's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X9,18,10,19 X17,22,18,9 X21,1,22,8 X20,15,21,16 X7,16,8,17 X4,13,5,14 X6,20,7,19
Gauss code	{1, -2, 3, -10, 4, -11, -9, 7}, {-5, -1, 2, -3, 10, -4, 8, 9, -6, 5, 11, -8, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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