L10a48

L10a47

L10a49

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Link Presentations

[edit Notes on L10a48's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X20,8,5,7 X18,10,19,9 X16,12,17,11 X12,16,13,15 X10,18,11,17 X8,20,9,19 X2536 X4,13,1,14
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 4, -7, 5, -6, 10, -2, 6, -5, 7, -4, 8, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {3t(2)t(1)-4t(1)-4t(2)+3}{{\sqrt {t(1)}}{\sqrt {t(2)}}}}$ (db)
Jones polynomial	$-q^{11/2}+2q^{9/2}-2q^{7/2}+3q^{5/2}-4q^{3/2}+4{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {3}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {1}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}-za^{-5}+z^{3}a^{-3}-2a^{3}z-a^{3}z^{-1}+za^{-3}+az^{3}+z^{3}a^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-3z^{8}a^{-2}-2z^{8}a^{-4}-z^{8}-az^{7}+4z^{7}a^{-1}+4z^{7}a^{-3}-z^{7}a^{-5}-a^{2}z^{6}+14z^{6}a^{-2}+11z^{6}a^{-4}+2z^{6}-a^{3}z^{5}+az^{5}-6z^{5}a^{-1}-3z^{5}a^{-3}+5z^{5}a^{-5}-a^{4}z^{4}-18z^{4}a^{-2}-17z^{4}a^{-4}-a^{5}z^{3}-a^{3}z^{3}+6z^{3}a^{-1}-6z^{3}a^{-5}+7z^{2}a^{-2}+7z^{2}a^{-4}+2a^{5}z+2a^{3}z-za^{-1}+za^{-5}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                     1 1 10                   1   -1 8                 1 1   0 6               2 1     -1 4             2 1       1 2           2 2         0 0         2 2           0 -2       2 3             1 -4     1 1               0 -6     2                 2 -8 1 1                   0 -10 1                     1

Integral Khovanov Homology (db, data source)

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

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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