L11a324

L11a323

L11a325

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Link Presentations

[edit Notes on L11a324's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X16,10,17,9 X6,13,7,14 X14,7,15,8 X8,15,1,16 X22,18,9,17 X4,20,5,19 X20,6,21,5 X18,22,19,21
Gauss code	{1, -2, 3, -9, 10, -5, 6, -7}, {4, -1, 2, -3, 5, -6, 7, -4, 8, -11, 9, -10, 11, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{3}v^{3}-2u^{3}v^{2}-2u^{2}v^{3}+6u^{2}v^{2}-3u^{2}v-3uv^{2}+6uv-2u-2v+2}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{7/2}-2q^{5/2}+4q^{3/2}-7{\sqrt {q}}+{\frac {7}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {4}{q^{11/2}}}+{\frac {2}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{3}z^{7}-az^{7}+a^{5}z^{5}-5a^{3}z^{5}-5az^{5}+z^{5}a^{-1}+4a^{5}z^{3}-8a^{3}z^{3}-8az^{3}+4z^{3}a^{-1}+4a^{5}z-5a^{3}z-5az+4za^{-1}+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{3}-a^{9}z+2a^{8}z^{4}-a^{8}z^{2}+3a^{7}z^{5}-2a^{7}z^{3}+a^{7}z+4a^{6}z^{6}-5a^{6}z^{4}+2a^{6}z^{2}+5a^{5}z^{7}-11a^{5}z^{5}+8a^{5}z^{3}-5a^{5}z+a^{5}z^{-1}+5a^{4}z^{8}-14a^{4}z^{6}+10a^{4}z^{4}-3a^{4}z^{2}-a^{4}+3a^{3}z^{9}-5a^{3}z^{7}-11a^{3}z^{5}+18a^{3}z^{3}-8a^{3}z+a^{3}z^{-1}+a^{2}z^{10}+3a^{2}z^{8}+z^{8}a^{-2}-24a^{2}z^{6}-6z^{6}a^{-2}+29a^{2}z^{4}+12z^{4}a^{-2}-9a^{2}z^{2}-8z^{2}a^{-2}+5az^{9}+2z^{9}a^{-1}-21az^{7}-11z^{7}a^{-1}+23az^{5}+20z^{5}a^{-1}-8az^{3}-15z^{3}a^{-1}+4az+5za^{-1}+z^{10}-z^{8}-12z^{6}+24z^{4}-11z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 8                       1 -1 6                     1   1 4                   3 1   -2 2                 4 1     3 0               3 3       0 -2             6 4         2 -4           4 4           0 -6         4 5             -1 -8       2 4               2 -10     2 4                 -2 -12   1 3                   2 -14   1                     -1 -16 1                       1

Integral Khovanov Homology (db, data source)

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

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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