L11n169

L11n168

L11n170

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Link Presentations

[edit Notes on L11n169's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X15,20,16,21 X11,18,12,19 X19,12,20,13 X17,22,18,7 X21,16,22,17 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, -6, 7, 11, -4, -5, 9, -8, 6, -7, 5, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{6}-u^{2}v^{4}+u^{2}v^{3}+2uv^{4}-3uv^{3}+2uv^{2}+v^{3}-v^{2}+1}{uv^{3}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}+{\frac {1}{q^{29/2}}}-{\frac {2}{q^{27/2}}}+{\frac {3}{q^{25/2}}}-{\frac {4}{q^{23/2}}}+{\frac {4}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{15/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-2z^{3}a^{13}-5za^{13}-2a^{13}z^{-1}+z^{7}a^{11}+9z^{5}a^{11}+24z^{3}a^{11}+22za^{11}+5a^{11}z^{-1}-z^{9}a^{9}-9z^{7}a^{9}-28z^{5}a^{9}-37z^{3}a^{9}-20za^{9}-3a^{9}z^{-1}$ (db)
Kauffman polynomial	$a^{18}z^{4}-2a^{18}z^{2}+2a^{17}z^{5}-4a^{17}z^{3}+a^{17}z+2a^{16}z^{6}-4a^{16}z^{4}+3a^{16}z^{2}-a^{16}+a^{15}z^{7}-a^{15}z^{5}+a^{15}z^{3}+2a^{14}z^{6}-3a^{14}z^{4}+3a^{14}z^{2}+2a^{13}z^{5}-3a^{13}z^{3}+5a^{13}z-2a^{13}z^{-1}+a^{12}z^{8}-9a^{12}z^{6}+26a^{12}z^{4}-24a^{12}z^{2}+5a^{12}+a^{11}z^{9}-10a^{11}z^{7}+33a^{11}z^{5}-45a^{11}z^{3}+26a^{11}z-5a^{11}z^{-1}+a^{10}z^{8}-9a^{10}z^{6}+24a^{10}z^{4}-22a^{10}z^{2}+5a^{10}+a^{9}z^{9}-9a^{9}z^{7}+28a^{9}z^{5}-37a^{9}z^{3}+20a^{9}z-3a^{9}z^{-1}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -8                       1 1 -10                       1 1 -12                   1     1 -14               2         2 -16             1 1 1       -1 -18           3 2           1 -20         2 2 1           -1 -22       2 2               0 -24     1 2                 1 -26   1 2                   -1 -28   1                     1 -30 1                       -1

Integral Khovanov Homology (db, data source)

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

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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