L10a79

L10a78

L10a80

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Link Presentations

[edit Notes on L10a79's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{2}v^{3}-5u^{2}v^{2}+4u^{2}v-u^{2}+uv^{4}-5uv^{3}+9uv^{2}-5uv+u-v^{4}+4v^{3}-5v^{2}+2v}{uv^{2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+8{\sqrt {q}}-{\frac {12}{\sqrt {q}}}+{\frac {14}{q^{3/2}}}-{\frac {16}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {11}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+5za^{5}+3a^{5}z^{-1}-2z^{5}a^{3}-5z^{3}a^{3}-6za^{3}-2a^{3}z^{-1}-z^{5}a+za+z^{3}a^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+3a^{8}z^{2}-a^{8}+3a^{7}z^{7}-8a^{7}z^{5}+7a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+4a^{6}z^{8}-7a^{6}z^{6}-2a^{6}z^{4}+7a^{6}z^{2}-3a^{6}+2a^{5}z^{9}+6a^{5}z^{7}-26a^{5}z^{5}+24a^{5}z^{3}-10a^{5}z+3a^{5}z^{-1}+11a^{4}z^{8}-20a^{4}z^{6}+3a^{4}z^{4}+6a^{4}z^{2}-3a^{4}+2a^{3}z^{9}+13a^{3}z^{7}-36a^{3}z^{5}+27a^{3}z^{3}-9a^{3}z+2a^{3}z^{-1}+7a^{2}z^{8}-4a^{2}z^{6}-7a^{2}z^{4}+z^{4}a^{-2}+4a^{2}z^{2}+10az^{7}-14az^{5}+4z^{5}a^{-1}+8az^{3}-2z^{3}a^{-1}-2az+8z^{6}-8z^{4}+2z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                     1 -1 4                   3   3 2                 5 1   -4 0               7 3     4 -2             8 6       -2 -4           8 6         2 -6         6 9           3 -8       5 7             -2 -10     2 6               4 -12   1 5                 -4 -14   2                   2 -16 1                     -1

Integral Khovanov Homology (db, data source)

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

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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