L11n91

L11n90

L11n92

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Link Presentations

[edit Notes on L11n91's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,16,8,17 X11,20,12,21 X17,22,18,5 X21,18,22,19 X19,10,20,11 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, -8, 7, -4, -2, 11, 8, -9, 3, -5, 6, -7, 4, -6, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {2}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {2}{q^{11/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}(-z)-a^{9}z^{-1}+a^{7}z^{3}+3a^{7}z+2a^{7}z^{-1}-a^{5}z^{3}-2a^{5}z-a^{5}z^{-1}+a^{3}z^{5}+3a^{3}z^{3}+2a^{3}z+a^{3}z^{-1}-az^{3}-2az-az^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{10}+7z^{6}a^{10}-15z^{4}a^{10}+12z^{2}a^{10}-3a^{10}-z^{9}a^{9}+7z^{7}a^{9}-14z^{5}a^{9}+10z^{3}a^{9}-3za^{9}+a^{9}z^{-1}-2z^{8}a^{8}+16z^{6}a^{8}-36z^{4}a^{8}+27z^{2}a^{8}-7a^{8}-z^{9}a^{7}+8z^{7}a^{7}-18z^{5}a^{7}+14z^{3}a^{7}-8za^{7}+2a^{7}z^{-1}-z^{8}a^{6}+8z^{6}a^{6}-20z^{4}a^{6}+15z^{2}a^{6}-4a^{6}-4z^{5}a^{5}+10z^{3}a^{5}-7za^{5}+a^{5}z^{-1}-3z^{6}a^{4}+6z^{4}a^{4}-z^{2}a^{4}-z^{7}a^{3}-z^{5}a^{3}+9z^{3}a^{3}-5za^{3}+a^{3}z^{-1}-2z^{6}a^{2}+5z^{4}a^{2}-z^{2}a^{2}-a^{2}-z^{5}a+3z^{3}a-3za+az^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     1   -1 -2                   3 1   2 -4                 2 3     1 -6             1 3 1       1 -8             1 2         1 -10         1 3 3           -1 -12       1   1             2 -14       1 2               -1 -16   1 1                   0 -18                         0 -20 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $i=0$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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