L11a169

L11a168

L11a170

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Link Presentations

[edit Notes on L11a169's Link Presentations]

Planar diagram presentation	X8192 X20,11,21,12 X10,4,11,3 X2,17,3,18 X14,5,15,6 X6718 X16,10,17,9 X18,13,19,14 X22,16,7,15 X12,19,13,20 X4,22,5,21
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, 2, -10, 8, -5, 9, -7, 4, -8, 10, -2, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{4}-6u^{2}v^{3}+6u^{2}v^{2}-2u^{2}v-2uv^{4}+9uv^{3}-15uv^{2}+9uv-2u-2v^{3}+6v^{2}-6v+2}{uv^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+14{\sqrt {q}}-{\frac {20}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {22}{q^{5/2}}}+{\frac {19}{q^{7/2}}}-{\frac {14}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-2a^{5}z^{3}+a^{5}z^{-1}+a^{3}z^{7}+3a^{3}z^{5}+2a^{3}z^{3}-2a^{3}z-2a^{3}z^{-1}+az^{7}+3az^{5}-z^{5}a^{-1}+3az^{3}-2z^{3}a^{-1}+2az-za^{-1}+2az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-3a^{4}z^{10}-3a^{2}z^{10}-7a^{5}z^{9}-15a^{3}z^{9}-8az^{9}-7a^{6}z^{8}-6a^{4}z^{8}-9a^{2}z^{8}-10z^{8}-4a^{7}z^{7}+15a^{5}z^{7}+38a^{3}z^{7}+11az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+18a^{6}z^{6}+31a^{4}z^{6}+34a^{2}z^{6}-4z^{6}a^{-2}+18z^{6}+10a^{7}z^{5}-8a^{5}z^{5}-35a^{3}z^{5}-3az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-13a^{6}z^{4}-35a^{4}z^{4}-38a^{2}z^{4}+5z^{4}a^{-2}-13z^{4}-5a^{7}z^{3}+6a^{3}z^{3}-6az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}+4a^{6}z^{2}+13a^{4}z^{2}+14a^{2}z^{2}+5z^{2}+2a^{5}z+6a^{3}z+6az+2za^{-1}-a^{2}-a^{5}z^{-1}-2a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 8 3     -5 0               12 6       6 -2             11 9         -2 -4           11 11           0 -6         9 12             3 -8       5 10               -5 -10     3 9                 6 -12   1 5                   -4 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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