L10a74

L10a73

L10a75

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Link Presentations

[edit Notes on L10a74's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X18,11,19,12 X20,13,7,14 X12,19,13,20 X14,17,15,18 X6718 X4,15,5,16
Gauss code	{1, -2, 3, -10, 4, -9}, {9, -1, 2, -3, 5, -7, 6, -8, 10, -4, 8, -5, 7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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