L11n165

L11n164

L11n166

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Link Presentations

[edit Notes on L11n165's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X15,20,16,21 X14,5,15,6 X4,13,5,14 X17,22,18,7 X21,16,22,17 X19,12,20,13 X11,18,12,19 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, -9, 8, 5, -4, -3, 7, -6, 9, -8, 3, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{5}-u^{2}v^{3}+u^{2}v^{2}+uv^{6}-uv^{5}+uv^{3}-uv+u+v^{4}-v^{3}+v}{uv^{3}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{27/2}}}-{\frac {1}{q^{25/2}}}+{\frac {1}{q^{23/2}}}-{\frac {1}{q^{21/2}}}-{\frac {1}{q^{15/2}}}+{\frac {1}{q^{13/2}}}-{\frac {2}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-2za^{13}-2a^{13}z^{-1}+z^{5}a^{11}+7z^{3}a^{11}+11za^{11}+5a^{11}z^{-1}-z^{7}a^{9}-6z^{5}a^{9}-10z^{3}a^{9}-7za^{9}-3a^{9}z^{-1}-z^{7}a^{7}-6z^{5}a^{7}-10z^{3}a^{7}-5za^{7}$ (db)
Kauffman polynomial	$a^{16}z^{6}-5a^{16}z^{4}+6a^{16}z^{2}-a^{16}+a^{15}z^{7}-5a^{15}z^{5}+6a^{15}z^{3}-2a^{15}z+a^{14}z^{6}-5a^{14}z^{4}+4a^{14}z^{2}-3a^{13}z^{3}+4a^{13}z-2a^{13}z^{-1}+a^{12}z^{8}-8a^{12}z^{6}+19a^{12}z^{4}-19a^{12}z^{2}+5a^{12}+a^{11}z^{9}-8a^{11}z^{7}+22a^{11}z^{5}-31a^{11}z^{3}+21a^{11}z-5a^{11}z^{-1}+2a^{10}z^{8}-13a^{10}z^{6}+24a^{10}z^{4}-17a^{10}z^{2}+5a^{10}+a^{9}z^{9}-6a^{9}z^{7}+11a^{9}z^{5}-12a^{9}z^{3}+10a^{9}z-3a^{9}z^{-1}+a^{8}z^{8}-5a^{8}z^{6}+5a^{8}z^{4}+a^{7}z^{7}-6a^{7}z^{5}+10a^{7}z^{3}-5a^{7}z$ (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 -6                       1 1 -8                     1 1 0 -10                   1     1 -12               1 1 1     1 -14             1 2 1       0 -16           1 1 1         1 -18         1 3 2           0 -20       1                 1 -22       1 1               0 -24   1 1                   0 -26                         0 -28 1                       -1

Integral Khovanov Homology (db, data source)

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

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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