L11n54

L11n53

L11n55

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Link Presentations

[edit Notes on L11n54's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,17,12,16 X7,15,8,14 X15,9,16,8 X13,21,14,20 X17,5,18,22 X21,19,22,18 X19,13,20,12 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9, -6, 4, -5, 3, -7, 8, -9, 6, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(2)^{5}-2t(2)^{4}+t(1)t(2)^{3}+t(2)^{2}-2t(1)t(2)+2t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-q^{7/2}+2q^{5/2}-{\frac {1}{q^{5/2}}}-3q^{3/2}+{\frac {1}{q^{3/2}}}+q^{15/2}-q^{13/2}+2{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-1}-z^{5}a^{-3}+az^{3}-4z^{3}a^{-1}-4z^{3}a^{-3}+3az-4za^{-1}-3za^{-3}+za^{-5}+za^{-7}+2az^{-1}-2a^{-1}z^{-1}-a^{-3}z^{-1}+a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-3z^{8}a^{-2}-2z^{8}a^{-4}-z^{8}-az^{7}+4z^{7}a^{-1}+5z^{7}a^{-3}-z^{7}a^{-5}-z^{7}a^{-7}+16z^{6}a^{-2}+13z^{6}a^{-4}-z^{6}a^{-8}+4z^{6}+6az^{5}-z^{5}a^{-1}-6z^{5}a^{-3}+7z^{5}a^{-5}+6z^{5}a^{-7}-22z^{4}a^{-2}-24z^{4}a^{-4}+2z^{4}a^{-6}+5z^{4}a^{-8}-z^{4}-11az^{3}-5z^{3}a^{-1}+5z^{3}a^{-3}-9z^{3}a^{-5}-8z^{3}a^{-7}+9z^{2}a^{-2}+17z^{2}a^{-4}-2z^{2}a^{-6}-5z^{2}a^{-8}-5z^{2}+8az+5za^{-1}-4za^{-3}+za^{-5}+2za^{-7}-3a^{-4}+a^{-8}+3-2az^{-1}-2a^{-1}z^{-1}+a^{-3}z^{-1}+a^{-5}z^{-1}$ (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 7 χ 16                       1 -1 14                         0 12                 1 1 1   1 10               2 1       -1 8             1 2 1       0 6           3 3 1         -1 4         1 1 1           1 2       2 4 1             1 0     1 1 1               1 -2     2                   2 -4 1 1                     0 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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