L11n292

L11n291

L11n293

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Link Presentations

[edit Notes on L11n292's Link Presentations]

Planar diagram presentation	X6172 X11,18,12,19 X3849 X2,16,3,15 X16,7,17,8 X9,11,10,22 X17,1,18,4 X19,5,20,10 X5,12,6,13 X21,15,22,14 X13,21,14,20
Gauss code	{1, -4, -3, 7}, {-9, -1, 5, 3, -6, 8}, {-2, 9, -11, 10, 4, -5, -7, 2, -8, 11, -10, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(3)^{4}+t(1)t(2)t(3)^{4}-t(2)^{2}t(3)^{3}+2t(1)t(3)^{3}-2t(1)t(2)t(3)^{3}+t(2)t(3)^{3}+2t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}+t(1)t(2)t(3)^{2}-t(2)t(3)^{2}-2t(2)^{2}t(3)+t(1)t(3)-t(1)t(2)t(3)+2t(2)t(3)+t(2)^{2}-t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{5}-3q^{4}-q^{-4}+5q^{3}+3q^{-3}-6q^{2}-4q^{-2}+8q+7q^{-1}-6$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-4}+a^{-4}-a^{2}z^{4}-2z^{4}a^{-2}-2a^{2}z^{2}-5z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{-2}+z^{6}+4z^{4}+5z^{2}-2z^{-2}+1$ (db)
Kauffman polynomial	$2az^{9}+2z^{9}a^{-1}+3a^{2}z^{8}+5z^{8}a^{-2}+8z^{8}+a^{3}z^{7}-5az^{7}-2z^{7}a^{-1}+4z^{7}a^{-3}-14a^{2}z^{6}-21z^{6}a^{-2}+z^{6}a^{-4}-36z^{6}-4a^{3}z^{5}-6az^{5}-16z^{5}a^{-1}-14z^{5}a^{-3}+19a^{2}z^{4}+30z^{4}a^{-2}+z^{4}a^{-4}+48z^{4}+4a^{3}z^{3}+14az^{3}+24z^{3}a^{-1}+17z^{3}a^{-3}+3z^{3}a^{-5}-9a^{2}z^{2}-20z^{2}a^{-2}-3z^{2}a^{-4}+z^{2}a^{-6}-25z^{2}-a^{3}z-3az-7za^{-1}-7za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 11                   1 1 9                 2   -2 7               3 1   2 5             4 3     -1 3           4 3 1     2 1         4 6         2 -1       3 3 1         1 -3     2 5             3 -5   1 2               -1 -7   2                 2 -9 1                   -1

Integral Khovanov Homology (db, data source)

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

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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