L11n108

L11n107

L11n109

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Link Presentations

[edit Notes on L11n108's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,19,14,18 X17,11,18,10 X21,9,22,8 X7,17,8,16 X9,21,10,20 X15,5,16,22 X19,15,20,14 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -6, 5, -7, 4, 11, -2, -3, 9, -8, 6, -4, 3, -9, 7, -5, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-uv^{4}+3uv^{3}-2uv^{2}-uv+2u+2v^{5}-v^{4}-2v^{3}+3v^{2}-v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+3q^{7/2}-2q^{5/2}-{\frac {1}{q^{5/2}}}+q^{3/2}+{\frac {1}{q^{3/2}}}+q^{15/2}-2q^{13/2}+3q^{11/2}-{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-1}+az^{3}-5z^{3}a^{-1}-z^{3}a^{-3}-z^{3}a^{-5}+3az-5za^{-1}-za^{-3}+za^{-7}+2az^{-1}-2a^{-1}z^{-1}-a^{-3}z^{-1}+a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-2z^{8}a^{-2}-2z^{8}a^{-4}-z^{8}a^{-6}-z^{8}-az^{7}+6z^{7}a^{-1}+6z^{7}a^{-3}-3z^{7}a^{-5}-2z^{7}a^{-7}+14z^{6}a^{-2}+12z^{6}a^{-4}+2z^{6}a^{-6}-z^{6}a^{-8}+5z^{6}+6az^{5}-7z^{5}a^{-1}-8z^{5}a^{-3}+13z^{5}a^{-5}+8z^{5}a^{-7}-24z^{4}a^{-2}-21z^{4}a^{-4}+4z^{4}a^{-6}+4z^{4}a^{-8}-3z^{4}-10az^{3}-2z^{3}a^{-1}+3z^{3}a^{-3}-12z^{3}a^{-5}-7z^{3}a^{-7}+10z^{2}a^{-2}+15z^{2}a^{-4}-3z^{2}a^{-6}-4z^{2}a^{-8}-4z^{2}+7az+4za^{-1}-3za^{-3}+2za^{-5}+2za^{-7}-3a^{-4}+a^{-8}+3-2az^{-1}-2a^{-1}z^{-1}+a^{-3}z^{-1}+a^{-5}z^{-1}$ (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                     1   1 12                   2 1   -1 10               1 2 1     0 8               2 2       0 6           2 3 2         -1 4         1 2 2           1 2       1 3 2             0 0     1 1 3               3 -2     1                   1 -4 1 1                     0 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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