L11n30

L11n29

L11n31

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Link Presentations

[edit Notes on L11n30's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X19,1,20,4 X5,12,6,13 X3849 X9,16,10,17 X13,22,14,5 X15,10,16,11 X21,14,22,15 X11,20,12,21 X2,18,3,17
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -6, 8, -10, 4, -7, 9, -8, 6, 11, -2, -3, 10, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2(u-1)(v-1)}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{7/2}}}-{\frac {2}{q^{5/2}}}+{\frac {1}{q^{3/2}}}+{\frac {1}{q^{19/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{9}-a^{9}z^{-1}+z^{3}a^{7}+2za^{7}+a^{7}z^{-1}+z^{3}a^{5}+2za^{5}+2a^{5}z^{-1}-2za^{3}-2a^{3}z^{-1}-za$ (db)
Kauffman polynomial	$a^{10}z^{8}-7a^{10}z^{6}+16a^{10}z^{4}-14a^{10}z^{2}+4a^{10}+a^{9}z^{9}-6a^{9}z^{7}+10a^{9}z^{5}-4a^{9}z^{3}-a^{9}z^{-1}+3a^{8}z^{8}-19a^{8}z^{6}+38a^{8}z^{4}-30a^{8}z^{2}+9a^{8}+a^{7}z^{9}-4a^{7}z^{7}+8a^{7}z^{3}-3a^{7}z-a^{7}z^{-1}+2a^{6}z^{8}-11a^{6}z^{6}+18a^{6}z^{4}-13a^{6}z^{2}+4a^{6}+2a^{5}z^{7}-10a^{5}z^{5}+14a^{5}z^{3}-9a^{5}z+2a^{5}z^{-1}+a^{4}z^{6}-4a^{4}z^{4}+4a^{4}z^{2}-2a^{4}+2a^{3}z^{3}-5a^{3}z+2a^{3}z^{-1}+a^{2}z^{2}+az$ (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 0                   1 1 -2                 2 2 0 -4               1 1 1 1 -6             2 3     1 -8           2 1 1     2 -10         1 3 1       1 -12       2 2           0 -14       1 1           0 -16   1 2               -1 -18                     0 -20 1                   -1

Integral Khovanov Homology (db, data source)

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

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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