L10a78

L10a77

L10a79

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Link Presentations

[edit Notes on L10a78's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X6,13,1,14 X18,12,19,11 X16,5,17,6 X12,18,13,17 X20,16,7,15 X4,19,5,20
Gauss code	{1, -2, 3, -10, 7, -5}, {4, -1, 2, -3, 6, -8, 5, -4, 9, -7, 8, -6, 10, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}-4t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}+4t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+4t(2)^{2}+4t(1)t(2)-4t(2)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {8}{q^{9/2}}}-{\frac {11}{q^{7/2}}}-q^{5/2}+{\frac {10}{q^{5/2}}}+3q^{3/2}-{\frac {10}{q^{3/2}}}-{\frac {1}{q^{15/2}}}+{\frac {3}{q^{13/2}}}-{\frac {6}{q^{11/2}}}-5{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{5}+3a^{5}z^{3}+3a^{5}z+2a^{5}z^{-1}-a^{3}z^{7}-5a^{3}z^{5}-10a^{3}z^{3}-10a^{3}z-3a^{3}z^{-1}+2az^{5}+7az^{3}-z^{3}a^{-1}+6az-2za^{-1}+az^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{9}-3z^{4}a^{8}-6z^{5}a^{7}+5z^{3}a^{7}-2za^{7}-8z^{6}a^{6}+10z^{4}a^{6}-2z^{2}a^{6}-9z^{7}a^{5}+19z^{5}a^{5}-13z^{3}a^{5}+7za^{5}-2a^{5}z^{-1}-6z^{8}a^{4}+10z^{6}a^{4}+4z^{4}a^{4}-7z^{2}a^{4}+3a^{4}-2z^{9}a^{3}-5z^{7}a^{3}+31z^{5}a^{3}-31z^{3}a^{3}+12za^{3}-3a^{3}z^{-1}-9z^{8}a^{2}+31z^{6}a^{2}-26z^{4}a^{2}+z^{2}a^{2}+3a^{2}-2z^{9}a+3z^{7}a+10z^{5}a-17z^{3}a+5za-az^{-1}-3z^{8}+13z^{6}-17z^{4}+6z^{2}+1-z^{7}a^{-1}+4z^{5}a^{-1}-5z^{3}a^{-1}+2za^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                     1 1 4                   2   -2 2                 3 1   2 0               5 2     -3 -2             5 3       2 -4           6 6         0 -6         5 4           1 -8       3 6             3 -10     3 5               -2 -12     3                 3 -14 1 3                   -2 -16 1                     1

Integral Khovanov Homology (db, data source)

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

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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