L11n48

L11n47

L11n49

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Link Presentations

[edit Notes on L11n48's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X19,1,20,4 X5,14,6,15 X3849 X9,16,10,17 X15,10,16,11 X11,20,12,21 X13,22,14,5 X21,12,22,13 X2,18,3,17
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -6, 7, -8, 10, -9, 4, -7, 6, 11, -2, -3, 8, -10, 9}

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	$-{\frac {1}{q^{3/2}}}-{\frac {2}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {2}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)-3a^{9}z-2a^{9}z^{-1}+a^{7}z^{5}+5a^{7}z^{3}+8a^{7}z+4a^{7}z^{-1}-a^{5}z^{3}-2a^{5}z-a^{5}z^{-1}-a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{6}-5a^{12}z^{4}+6a^{12}z^{2}-2a^{12}+a^{11}z^{7}-4a^{11}z^{5}+2a^{11}z^{3}+a^{11}z+a^{10}z^{8}-4a^{10}z^{6}+2a^{10}z^{4}+2a^{10}z^{2}-a^{10}+a^{9}z^{9}-6a^{9}z^{7}+13a^{9}z^{5}-15a^{9}z^{3}+8a^{9}z-2a^{9}z^{-1}+2a^{8}z^{8}-12a^{8}z^{6}+24a^{8}z^{4}-20a^{8}z^{2}+6a^{8}+a^{7}z^{9}-7a^{7}z^{7}+18a^{7}z^{5}-22a^{7}z^{3}+15a^{7}z-4a^{7}z^{-1}+a^{6}z^{8}-7a^{6}z^{6}+17a^{6}z^{4}-15a^{6}z^{2}+5a^{6}+a^{5}z^{5}-4a^{5}z^{3}+5a^{5}z-a^{5}z^{-1}+a^{4}z^{2}-a^{4}+a^{3}z^{3}-3a^{3}z+a^{3}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   1 1 -4                 1 2 1 -6               1   1 2 -8             2 2     0 -10           2 1       1 -12         1 3 1       1 -14       2 2           0 -16       1 1           0 -18   1 2               -1 -20                     0 -22 1                   -1

Integral Khovanov Homology (db, data source)

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

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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