L11a272

L11a271

L11a273

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Link Presentations

[edit Notes on L11a272's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X14,6,15,5 X22,14,9,13 X18,8,19,7 X20,18,21,17 X16,22,17,21 X6,16,7,15 X8,20,1,19
Gauss code	{1, -2, 3, -4, 5, -10, 7, -11}, {4, -1, 2, -3, 6, -5, 10, -9, 8, -7, 11, -8, 9, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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