L11n164

L11n163

L11n165

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Link Presentations

[edit Notes on L11n164's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X12,17,13,18 X14,5,15,6 X4,13,5,14 X18,11,19,12 X19,7,20,22 X15,21,16,20 X21,17,22,16 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 6, -3, 5, -4, -8, 9, 3, -6, -7, 8, -9, 7}

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

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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