L10a72

L10a71

L10a73

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Link Presentations

[edit Notes on L10a72's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X18,11,19,12 X20,13,7,14 X12,19,13,20 X2738 X6,9,1,10
Gauss code	{1, -9, 2, -5, 4, -10}, {9, -1, 10, -2, 6, -8, 7, -3, 5, -4, 3, -6, 8, -7}

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

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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