L11n167

L11n166

L11n168

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Link Presentations

[edit Notes on L11n167's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X11,19,12,18 X22,19,7,20 X20,15,21,16 X16,21,17,22 X17,13,18,12 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, -5, 9, 11, -4, 7, -8, -9, 5, 6, -7, 8, -6}

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

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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