L11n392

L11n391

L11n393

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Link Presentations

[edit Notes on L11n392's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X8493 X22,14,19,13 X20,10,21,9 X10,20,11,19 X14,22,15,21 X11,18,12,5 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {8, -7, 9, -6}, {-4, -1, 2, -5, 7, -8, -10, 4, 6, -9, 11, -2, 3, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(3)-1)^{2}(t(3)t(2)-2t(2)-2t(3)+1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-5}+3q^{4}+4q^{-4}-6q^{3}-9q^{-3}+13q^{2}+13q^{-2}-14q-16q^{-1}+17$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{4}z^{2}+2a^{-4}z^{-2}+2a^{-4}+2a^{2}z^{4}+z^{4}a^{-2}+a^{2}z^{2}-3z^{2}a^{-2}-a^{2}z^{-2}-5a^{-2}z^{-2}-2a^{2}-8a^{-2}-z^{6}-z^{4}+3z^{2}+4z^{-2}+8$ (db)
Kauffman polynomial	$a^{5}z^{5}-a^{5}z^{3}+4a^{4}z^{6}-5a^{4}z^{4}+6z^{4}a^{-4}+2a^{4}z^{2}-11z^{2}a^{-4}-2a^{-4}z^{-2}+8a^{-4}+8a^{3}z^{7}+3z^{7}a^{-3}-14a^{3}z^{5}-3z^{5}a^{-3}+9a^{3}z^{3}+10z^{3}a^{-3}-3a^{3}z-13za^{-3}+a^{3}z^{-1}+5a^{-3}z^{-1}+7a^{2}z^{8}+5z^{8}a^{-2}-6a^{2}z^{6}-8z^{6}a^{-2}-4a^{2}z^{4}+17z^{4}a^{-2}-24z^{2}a^{-2}-a^{2}z^{-2}-5a^{-2}z^{-2}+2a^{2}+16a^{-2}+2az^{9}+2z^{9}a^{-1}+15az^{7}+10z^{7}a^{-1}-38az^{5}-26z^{5}a^{-1}+29az^{3}+29z^{3}a^{-1}-14az-24za^{-1}+5az^{-1}+9a^{-1}z^{-1}+12z^{8}-18z^{6}+12z^{4}-15z^{2}-4z^{-2}+11$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   3 3 7                 5 2 -3 5               8 1   7 3             6 5     -1 1           11 8       3 -1         7 8         1 -3       6 9           -3 -5     3 7             4 -7   1 6               -5 -9   3                 3 -11 1                   -1

Integral Khovanov Homology (db, data source)

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

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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