L11n49

L11n48

L11n50

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Link Presentations

[edit Notes on L11n49's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X5,14,6,15 X8493 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 $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{4}-v^{3}+v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {4}{q^{7/2}}}-{\frac {4}{q^{5/2}}}+{\frac {2}{q^{3/2}}}+{\frac {2}{q^{19/2}}}-{\frac {3}{q^{17/2}}}+{\frac {5}{q^{15/2}}}-{\frac {6}{q^{13/2}}}+{\frac {6}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}(-z)-2a^{9}z^{-1}-a^{7}z^{5}-2a^{7}z^{3}+3a^{7}z+4a^{7}z^{-1}+a^{5}z^{7}+5a^{5}z^{5}+7a^{5}z^{3}+2a^{5}z-a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-3z^{2}a^{12}+2a^{12}-z^{5}a^{11}-z^{3}a^{11}-za^{11}-2z^{6}a^{10}+2z^{4}a^{10}-3z^{2}a^{10}+a^{10}-2z^{7}a^{9}+z^{5}a^{9}+4z^{3}a^{9}-4za^{9}+2a^{9}z^{-1}-2z^{8}a^{8}+4z^{6}a^{8}-5z^{4}a^{8}+12z^{2}a^{8}-6a^{8}-z^{9}a^{7}+6z^{5}a^{7}-z^{3}a^{7}-5za^{7}+4a^{7}z^{-1}-4z^{8}a^{6}+15z^{6}a^{6}-17z^{4}a^{6}+12z^{2}a^{6}-5a^{6}-z^{9}a^{5}+z^{7}a^{5}+9z^{5}a^{5}-14z^{3}a^{5}+3za^{5}+a^{5}z^{-1}-2z^{8}a^{4}+9z^{6}a^{4}-10z^{4}a^{4}+a^{4}-z^{7}a^{3}+5z^{5}a^{3}-8z^{3}a^{3}+5za^{3}-a^{3}z^{-1}$ (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 3     0 -8           4 1       3 -10         2 3         1 -12       4 4           0 -14     1 2             1 -16   2 4               -2 -18   1                 1 -20 2                   -2

Integral Khovanov Homology (db, data source)

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

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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