L11n75

L11n74

L11n76

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Link Presentations

[edit Notes on L11n75's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(2)^{5}-t(2)^{4}+2t(1)t(2)^{3}-2t(2)^{3}-2t(1)t(2)^{2}+2t(2)^{2}-t(1)t(2)+2t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-2q^{9/2}+q^{7/2}-q^{5/2}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{3/2}}}+q^{15/2}-q^{13/2}+2q^{11/2}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}+3za^{-5}+2a^{-5}z^{-1}-z^{5}a^{-3}-6z^{3}a^{-3}-8za^{-3}-4a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-4z^{3}a^{-1}+3az-za^{-1}+az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-2z^{8}a^{-2}-z^{8}a^{-4}-z^{8}-az^{7}+6z^{7}a^{-1}+8z^{7}a^{-3}-z^{7}a^{-7}+14z^{6}a^{-2}+8z^{6}a^{-4}-2z^{6}a^{-6}-z^{6}a^{-8}+5z^{6}+6az^{5}-9z^{5}a^{-1}-21z^{5}a^{-3}-2z^{5}a^{-5}+4z^{5}a^{-7}-27z^{4}a^{-2}-20z^{4}a^{-4}+8z^{4}a^{-6}+5z^{4}a^{-8}-4z^{4}-10az^{3}+5z^{3}a^{-1}+27z^{3}a^{-3}+9z^{3}a^{-5}-3z^{3}a^{-7}+18z^{2}a^{-2}+21z^{2}a^{-4}-6z^{2}a^{-6}-7z^{2}a^{-8}-2z^{2}+6az-4za^{-1}-17za^{-3}-7za^{-5}-5a^{-2}-6a^{-4}+a^{-6}+2a^{-8}+1-az^{-1}+a^{-1}z^{-1}+4a^{-3}z^{-1}+2a^{-5}z^{-1}$ (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                   2 1   -1 10               1 1       0 8               1 2       1 6           2 3 1         0 4         1 1 1           1 2       1 3 2             0 0     1 1 2               2 -2     1                   1 -4 1 1                     0 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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