L11n432

L11n431

L11n433

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Link Presentations

[edit Notes on L11n432's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X15,20,16,21 X5,17,6,16 X11,19,12,18 X17,11,18,10 X19,5,20,4
Gauss code	{1, -4, 2, 11, -8, -6}, {4, -1, 5, 10, -9, -3}, {3, -2, -7, 8, -10, 9, -11, 7, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)t(2)+t(3))\left(-t(1)t(3)^{2}+t(1)t(2)t(3)^{2}-t(2)t(3)^{2}+t(1)+t(2)-1\right)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{9}-q^{8}+q^{7}+q^{5}+q^{4}+q^{2}-q+1$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-4}+z^{4}a^{-2}-5z^{4}a^{-4}+4z^{2}a^{-2}-5z^{2}a^{-4}+z^{2}a^{-8}+2a^{-2}-3a^{-6}+a^{-8}+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-10}-5z^{4}a^{-10}+5z^{2}a^{-10}+z^{7}a^{-9}-5z^{5}a^{-9}+5z^{3}a^{-9}+z^{8}a^{-8}-7z^{6}a^{-8}+16z^{4}a^{-8}-17z^{2}a^{-8}-a^{-8}z^{-2}+7a^{-8}+z^{7}a^{-7}-7z^{5}a^{-7}+13z^{3}a^{-7}-9za^{-7}+2a^{-7}z^{-1}+z^{8}a^{-6}-8z^{6}a^{-6}+21z^{4}a^{-6}-25z^{2}a^{-6}-2a^{-6}z^{-2}+11a^{-6}+z^{7}a^{-5}-7z^{5}a^{-5}+13z^{3}a^{-5}-9za^{-5}+2a^{-5}z^{-1}+z^{6}a^{-4}-5z^{4}a^{-4}+3z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+z^{7}a^{-3}-5z^{5}a^{-3}+5z^{3}a^{-3}+z^{6}a^{-2}-5z^{4}a^{-2}+6z^{2}a^{-2}-2a^{-2}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 19                     1 1 17                   1 1 0 15               1 1     0 13             1 1 1     1 11           1 3 1       1 9         2 1 1         2 7       1 4 2           1 5     1 1 1             1 3   1 2 1               0 1                       0 -1 1                     1

Integral Khovanov Homology (db, data source)

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

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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