L11n31

L11n30

L11n32

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Link Presentations

[edit Notes on L11n31's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X5,12,6,13 X8493 X9,16,10,17 X13,22,14,5 X15,10,16,11 X21,14,22,15 X11,20,12,21 X2,18,3,17
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 8, -10, 4, -7, 9, -8, 6, 11, -2, 3, 10, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {10}{q^{7/2}}}+{\frac {9}{q^{9/2}}}-{\frac {9}{q^{11/2}}}+{\frac {6}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {2}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}-z^{3}a^{7}+a^{7}z^{-1}+z^{5}a^{5}+2z^{3}a^{5}+3za^{5}+2a^{5}z^{-1}+z^{5}a^{3}+z^{3}a^{3}-2za^{3}-2a^{3}z^{-1}-z^{3}a-za$ (db)
Kauffman polynomial	$3a^{10}z^{4}-8a^{10}z^{2}+4a^{10}+a^{9}z^{7}-a^{9}z^{5}+a^{9}z^{3}-2a^{9}z-a^{9}z^{-1}+2a^{8}z^{8}-7a^{8}z^{6}+18a^{8}z^{4}-21a^{8}z^{2}+9a^{8}+a^{7}z^{9}+a^{7}z^{7}-8a^{7}z^{5}+17a^{7}z^{3}-7a^{7}z-a^{7}z^{-1}+5a^{6}z^{8}-14a^{6}z^{6}+20a^{6}z^{4}-11a^{6}z^{2}+4a^{6}+a^{5}z^{9}+4a^{5}z^{7}-16a^{5}z^{5}+22a^{5}z^{3}-11a^{5}z+2a^{5}z^{-1}+3a^{4}z^{8}-4a^{4}z^{6}-a^{4}z^{4}+3a^{4}z^{2}-2a^{4}+4a^{3}z^{7}-8a^{3}z^{5}+4a^{3}z^{3}-5a^{3}z+2a^{3}z^{-1}+3a^{2}z^{6}-6a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                   1 1 0                 2   -2 -2               4 1   3 -4             5 4     -1 -6           5 2       3 -8         4 5         1 -10       5 5           0 -12     1 4             3 -14   2 5               -3 -16   1                 1 -18 2                   -2

Integral Khovanov Homology (db, data source)

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

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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