L11a165

L11a164

L11a166

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Link Presentations

[edit Notes on L11a165's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{2}v^{4}-4u^{2}v^{3}+3u^{2}v^{2}+3uv^{3}-5uv^{2}+3uv+3v^{2}-4v+2}{uv^{2}}}$ (db)
Jones polynomial	${\frac {6}{q^{9/2}}}+q^{7/2}-{\frac {8}{q^{7/2}}}-2q^{5/2}+{\frac {8}{q^{5/2}}}+4q^{3/2}-{\frac {9}{q^{3/2}}}-{\frac {1}{q^{15/2}}}+{\frac {2}{q^{13/2}}}-{\frac {4}{q^{11/2}}}-6{\sqrt {q}}+{\frac {7}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{5}+4a^{5}z^{3}+4a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-5a^{3}z^{5}-8a^{3}z^{3}-5a^{3}z-az^{7}-5az^{5}+z^{5}a^{-1}-8az^{3}+4z^{3}a^{-1}-6az+4za^{-1}-2az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-5az^{9}-2z^{9}a^{-1}-4a^{4}z^{8}-2a^{2}z^{8}-z^{8}a^{-2}+z^{8}-5a^{5}z^{7}+7a^{3}z^{7}+23az^{7}+11z^{7}a^{-1}-4a^{6}z^{6}+8a^{4}z^{6}+20a^{2}z^{6}+6z^{6}a^{-2}+14z^{6}-3a^{7}z^{5}+12a^{5}z^{5}+2a^{3}z^{5}-32az^{5}-19z^{5}a^{-1}-2a^{8}z^{4}+5a^{6}z^{4}+2a^{4}z^{4}-26a^{2}z^{4}-12z^{4}a^{-2}-33z^{4}-a^{9}z^{3}+2a^{7}z^{3}-11a^{5}z^{3}-7a^{3}z^{3}+19az^{3}+12z^{3}a^{-1}+a^{8}z^{2}-2a^{6}z^{2}-5a^{4}z^{2}+12a^{2}z^{2}+9z^{2}a^{-2}+23z^{2}+a^{9}z-a^{7}z+6a^{5}z+4a^{3}z-8az-4za^{-1}+a^{4}-3a^{2}-2a^{-2}-5-a^{5}z^{-1}+2az^{-1}+a^{-1}z^{-1}$ (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 4 5 χ 8                       1 -1 6                     1   1 4                   3 1   -2 2                 3 1     2 0               4 3       -1 -2             5 3         2 -4           4 5           1 -6         4 4             0 -8       2 4               2 -10     2 4                 -2 -12     2                   2 -14 1 2                     -1 -16 1                       1

Integral Khovanov Homology (db, data source)

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

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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