L11n194

L11n193

L11n195

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Link Presentations

[edit Notes on L11n194's Link Presentations]

Planar diagram presentation	X8192 X16,6,17,5 X9,20,10,21 X21,10,22,11 X18,16,19,15 X14,20,15,19 X2,11,3,12 X12,3,13,4 X4758 X22,14,7,13 X6,18,1,17
Gauss code	{1, -7, 8, -9, 2, -11}, {9, -1, -3, 4, 7, -8, 10, -6, 5, -2, 11, -5, 6, 3, -4, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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