L11n42

L11n41

L11n43

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Link Presentations

[edit Notes on L11n42's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X9,18,10,19 X3849 X5,14,6,15 X15,22,16,5 X17,20,18,21 X21,16,22,17 X19,10,20,11 X11,2,12,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 10, -11, -2, 3, 6, -7, 9, -8, 4, -10, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3t(1)t(2)^{3}-4t(1)t(2)^{2}+2t(2)^{2}+2t(1)t(2)-4t(2)+3}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {6}{q^{13/2}}}+{\frac {6}{q^{15/2}}}-{\frac {6}{q^{17/2}}}+{\frac {4}{q^{19/2}}}-{\frac {3}{q^{21/2}}}+{\frac {2}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+z^{3}a^{11}+3za^{11}+2a^{11}z^{-1}-z^{5}a^{9}-3z^{3}a^{9}-2za^{9}-a^{9}z^{-1}-z^{5}a^{7}-2z^{3}a^{7}+za^{7}+a^{7}z^{-1}-z^{5}a^{5}-4z^{3}a^{5}-4za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$3a^{14}z^{4}-8a^{14}z^{2}+3a^{14}+a^{13}z^{7}-a^{13}z^{5}-3a^{13}z^{3}+3a^{13}z-a^{13}z^{-1}+2a^{12}z^{8}-10a^{12}z^{6}+24a^{12}z^{4}-24a^{12}z^{2}+7a^{12}+a^{11}z^{9}-4a^{11}z^{7}+10a^{11}z^{5}-11a^{11}z^{3}+8a^{11}z-2a^{11}z^{-1}+3a^{10}z^{8}-13a^{10}z^{6}+25a^{10}z^{4}-16a^{10}z^{2}+4a^{10}+a^{9}z^{9}-4a^{9}z^{7}+9a^{9}z^{5}-7a^{9}z^{3}+4a^{9}z-a^{9}z^{-1}+a^{8}z^{8}-2a^{8}z^{6}+2a^{8}z^{4}-a^{8}z^{2}+a^{7}z^{7}-a^{7}z^{5}-3a^{7}z^{3}+3a^{7}z-a^{7}z^{-1}+a^{6}z^{6}-2a^{6}z^{4}-a^{6}z^{2}+a^{6}+a^{5}z^{5}-4a^{5}z^{3}+4a^{5}z-a^{5}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   1 1 -6                 1 1 0 -8               2     2 -10             2 1     -1 -12           4 2       2 -14         3 3         0 -16       3 3           0 -18     1 3             2 -20   2 3               -1 -22   1                 1 -24 2                   -2

Integral Khovanov Homology (db, data source)

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

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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