L11n35

L11n34

L11n36

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Link Presentations

[edit Notes on L11n35's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X9,14,10,15 X8493 X12,5,13,6 X22,13,5,14 X15,18,16,19 X11,17,12,16 X17,11,18,10 X2,20,3,19
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, -4, 10, -9, -6, 7, 4, -8, 9, -10, 8, 11, -2, 3, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-2)(t(2)-1)(2t(2)-1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {10}{q^{9/2}}}+{\frac {12}{q^{7/2}}}-{\frac {12}{q^{5/2}}}-2q^{3/2}+{\frac {11}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {7}{q^{11/2}}}+4{\sqrt {q}}-{\frac {9}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+3a^{5}z^{3}+4a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-6a^{3}z^{3}-8a^{3}z-3a^{3}z^{-1}+4az^{3}+7az+4az^{-1}-2za^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{5}z^{9}-2a^{3}z^{9}-5a^{6}z^{8}-10a^{4}z^{8}-5a^{2}z^{8}-4a^{7}z^{7}-5a^{5}z^{7}-5a^{3}z^{7}-4az^{7}-a^{8}z^{6}+12a^{6}z^{6}+26a^{4}z^{6}+12a^{2}z^{6}-z^{6}+11a^{7}z^{5}+27a^{5}z^{5}+26a^{3}z^{5}+10az^{5}+2a^{8}z^{4}-4a^{6}z^{4}-17a^{4}z^{4}-13a^{2}z^{4}-2z^{4}-7a^{7}z^{3}-22a^{5}z^{3}-33a^{3}z^{3}-21az^{3}-3z^{3}a^{-1}-a^{8}z^{2}-a^{2}z^{2}+a^{7}z+6a^{5}z+15a^{3}z+15az+5za^{-1}+a^{6}+3a^{4}+3a^{2}+2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 4                   2 2 2                 2   -2 0               7 2   5 -2             6 4     -2 -4           6 5       1 -6         6 6         0 -8       4 6           -2 -10     3 6             3 -12   1 4               -3 -14   3                 3 -16 1                   -1

Integral Khovanov Homology (db, data source)

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

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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