L11n179

L11n178

L11n180

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Link Presentations

[edit Notes on L11n179's Link Presentations]

Planar diagram presentation	X8192 X3,10,4,11 X16,7,17,8 X20,10,21,9 X22,15,7,16 X14,6,15,5 X18,14,19,13 X12,20,13,19 X11,4,12,5 X6,18,1,17 X2,21,3,22
Gauss code	{1, -11, -2, 9, 6, -10}, {3, -1, 4, 2, -9, -8, 7, -6, 5, -3, 10, -7, 8, -4, 11, -5}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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