L11n211

L11n210

L11n212

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Link Presentations

[edit Notes on L11n211's Link Presentations]

Planar diagram presentation	X10,1,11,2 X15,21,16,20 X5,14,6,15 X3,12,4,13 X13,4,14,5 X2,19,3,20 X16,7,17,8 X8,9,1,10 X18,12,19,11 X22,18,9,17 X21,6,22,7
Gauss code	{1, -6, -4, 5, -3, 11, 7, -8}, {8, -1, 9, 4, -5, 3, -2, -7, 10, -9, 6, 2, -11, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{3}v^{2}+u^{2}v^{5}-2u^{2}v^{4}+3u^{2}v^{3}-3u^{2}v^{2}+2u^{2}v+2uv^{4}-3uv^{3}+3uv^{2}-2uv+u+v^{3}}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	${\frac {8}{q^{9/2}}}-{\frac {8}{q^{7/2}}}+{\frac {5}{q^{5/2}}}-{\frac {3}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {2}{q^{17/2}}}-{\frac {5}{q^{15/2}}}+{\frac {7}{q^{13/2}}}-{\frac {8}{q^{11/2}}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+4z^{3}a^{7}+6za^{7}+2a^{7}z^{-1}-z^{7}a^{5}-6z^{5}a^{5}-14z^{3}a^{5}-13za^{5}-3a^{5}z^{-1}+z^{5}a^{3}+3z^{3}a^{3}+3za^{3}+a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-2z^{6}a^{10}+4z^{4}a^{10}-z^{2}a^{10}-3z^{7}a^{9}+7z^{5}a^{9}-6z^{3}a^{9}+3za^{9}-2z^{8}a^{8}+2z^{6}a^{8}+z^{2}a^{8}-z^{9}a^{7}+z^{7}a^{7}-6z^{5}a^{7}+11z^{3}a^{7}-8za^{7}+2a^{7}z^{-1}-3z^{8}a^{6}+7z^{6}a^{6}-14z^{4}a^{6}+12z^{2}a^{6}-3a^{6}-z^{9}a^{5}+4z^{7}a^{5}-17z^{5}a^{5}+25z^{3}a^{5}-16za^{5}+3a^{5}z^{-1}-z^{8}a^{4}+3z^{6}a^{4}-11z^{4}a^{4}+12z^{2}a^{4}-3a^{4}-3z^{5}a^{3}+5z^{3}a^{3}-3za^{3}+a^{3}z^{-1}-z^{4}a^{2}+2z^{2}a^{2}-a^{2}$ (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 0                   1 -1 -2                 2   2 -4               4 2   -2 -6             4 1     3 -8           4 4       0 -10         4 4         0 -12       3 4           1 -14     2 4             -2 -16     3               3 -18 1 2                 -1 -20 1                   1

Integral Khovanov Homology (db, data source)

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

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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