L11n185

L11n184

L11n186

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Link Presentations

[edit Notes on L11n185's Link Presentations]

Planar diagram presentation	X8192 X9,21,10,20 X21,1,22,6 X18,8,19,7 X3,10,4,11 X15,12,16,13 X5,14,6,15 X13,4,14,5 X11,16,12,17 X22,18,7,17 X2,20,3,19
Gauss code	{1, -11, -5, 8, -7, 3}, {4, -1, -2, 5, -9, 6, -8, 7, -6, 9, 10, -4, 11, 2, -3, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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