L10a73

L10a72

L10a74

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Link Presentations

[edit Notes on L10a73's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X20,11,7,12 X18,13,19,14 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, -7, 8, -6}

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

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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