L11n7

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Link Presentations

[edit Notes on L11n7's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X3849 X5,11,6,10 X11,20,12,21 X19,22,20,5 X13,19,14,18 X21,12,22,13 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 10, -9, 4, -11, -2, 3, 9, -8, 7, -10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{5}-4t(1)t(2)^{4}+4t(1)t(2)^{3}-2t(2)^{3}-2t(1)t(2)^{2}+4t(2)^{2}-4t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {7}{q^{13/2}}}+{\frac {5}{q^{15/2}}}-{\frac {4}{q^{17/2}}}+{\frac {2}{q^{19/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+z^{3}a^{9}+4za^{9}+3a^{9}z^{-1}-2z^{5}a^{7}-8z^{3}a^{7}-8za^{7}-3a^{7}z^{-1}+z^{7}a^{5}+5z^{5}a^{5}+8z^{3}a^{5}+6za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-4za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-3z^{2}a^{12}+a^{12}-z^{5}a^{11}-3z^{3}a^{11}+3za^{11}-a^{11}z^{-1}-3z^{6}a^{10}+5z^{4}a^{10}-6z^{2}a^{10}+2a^{10}-4z^{7}a^{9}+11z^{5}a^{9}-15z^{3}a^{9}+12za^{9}-3a^{9}z^{-1}-3z^{8}a^{8}+7z^{6}a^{8}-4z^{4}a^{8}+3z^{2}a^{8}-z^{9}a^{7}-3z^{7}a^{7}+21z^{5}a^{7}-27z^{3}a^{7}+15za^{7}-3a^{7}z^{-1}-5z^{8}a^{6}+19z^{6}a^{6}-20z^{4}a^{6}+9z^{2}a^{6}-2a^{6}-z^{9}a^{5}+14z^{5}a^{5}-23z^{3}a^{5}+11za^{5}-2a^{5}z^{-1}-2z^{8}a^{4}+9z^{6}a^{4}-11z^{4}a^{4}+3z^{2}a^{4}-z^{7}a^{3}+5z^{5}a^{3}-8z^{3}a^{3}+5za^{3}-a^{3}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                   1 1 -2                 1   -1 -4               3 1   2 -6             3 2     -1 -8           4 2       2 -10         3 3         0 -12       4 4           0 -14     2 4             2 -16   2 3               -1 -18   2                 2 -20 2                   -2

Integral Khovanov Homology (db, data source)

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

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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