L11n69

L11n68

L11n70

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Link Presentations

[edit Notes on L11n69's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,16,8,17 X22,18,5,17 X18,12,19,11 X20,14,21,13 X12,20,13,19 X14,22,15,21 X15,8,16,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -7, 6, -8, -9, 3, 4, -5, 7, -6, 8, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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