L11a52

L11a51

L11a53

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Link Presentations

[edit Notes on L11a52's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X14,6,15,5 X10,4,11,3 X20,12,21,11 X22,14,5,13 X12,22,13,21 X16,9,17,10 X2,16,3,15 X8,17,9,18
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 6, -8, 7, -4, 10, -9, 11, -2, 3, -6, 8, -7}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2(t(1)-1)(t(2)-1)\left(t(2)^{4}-t(2)^{3}+t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{15/2}+3q^{13/2}-6q^{11/2}+9q^{9/2}-11q^{7/2}+12q^{5/2}-12q^{3/2}+10{\sqrt {q}}-{\frac {8}{\sqrt {q}}}+{\frac {4}{q^{3/2}}}-{\frac {3}{q^{5/2}}}+{\frac {1}{q^{7/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-5}-3z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+5z^{3}a^{-3}+5za^{-3}+3a^{-3}z^{-1}+z^{7}a^{-1}-az^{5}+4z^{5}a^{-1}-3az^{3}+3z^{3}a^{-1}-3za^{-1}+2az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}+6z^{5}a^{-7}-4z^{3}a^{-7}+za^{-7}+9z^{6}a^{-6}-13z^{4}a^{-6}+5z^{2}a^{-6}-a^{-6}+10z^{7}a^{-5}-20z^{5}a^{-5}+9z^{3}a^{-5}-2za^{-5}+a^{-5}z^{-1}+8z^{8}a^{-4}-16z^{6}a^{-4}-z^{4}a^{-4}+8z^{2}a^{-4}-3a^{-4}+5z^{9}a^{-3}-9z^{7}a^{-3}-9z^{5}a^{-3}+16z^{3}a^{-3}-9za^{-3}+3a^{-3}z^{-1}+2z^{10}a^{-2}+a^{2}z^{8}-5a^{2}z^{6}-18z^{6}a^{-2}+7a^{2}z^{4}+17z^{4}a^{-2}-2a^{2}z^{2}+2z^{2}a^{-2}-3a^{-2}+3az^{9}+8z^{9}a^{-1}-17az^{7}-36z^{7}a^{-1}+31az^{5}+48z^{5}a^{-1}-18az^{3}-16z^{3}a^{-1}-az-7za^{-1}+2az^{-1}+4a^{-1}z^{-1}+2z^{10}-7z^{8}+2z^{6}+9z^{4}-3z^{2}-2$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                       1 1 14                     2   -2 12                   4 1   3 10                 5 2     -3 8               6 4       2 6             6 5         -1 4           6 6           0 2         6 8             2 0       2 4               -2 -2     2 6                 4 -4   1 2                   -1 -6   2                     2 -8 1                       -1

Integral Khovanov Homology (db, data source)

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

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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