L11a285

L11a284

L11a286

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Link Presentations

[edit Notes on L11a285's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{4}-u^{2}v^{3}+u^{2}v^{2}+2uv^{3}-3uv^{2}+2uv+v^{2}-v+1\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+6q^{5/2}-11q^{3/2}+14{\sqrt {q}}-{\frac {17}{\sqrt {q}}}+{\frac {16}{q^{3/2}}}-{\frac {15}{q^{5/2}}}+{\frac {11}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{9}+a^{3}z^{7}-7az^{7}+z^{7}a^{-1}+5a^{3}z^{5}-18az^{5}+5z^{5}a^{-1}+8a^{3}z^{3}-19az^{3}+8z^{3}a^{-1}+3a^{3}z-6az+3za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-5a^{3}z^{9}-9az^{9}-4z^{9}a^{-1}-6a^{4}z^{8}-2a^{2}z^{8}-4z^{8}a^{-2}-5a^{5}z^{7}+11a^{3}z^{7}+28az^{7}+9z^{7}a^{-1}-3z^{7}a^{-3}-3a^{6}z^{6}+13a^{4}z^{6}+11a^{2}z^{6}+9z^{6}a^{-2}-z^{6}a^{-4}+5z^{6}-a^{7}z^{5}+10a^{5}z^{5}-15a^{3}z^{5}-44az^{5}-9z^{5}a^{-1}+9z^{5}a^{-3}+6a^{6}z^{4}-12a^{4}z^{4}-17a^{2}z^{4}-3z^{4}a^{-2}+3z^{4}a^{-4}-5z^{4}+2a^{7}z^{3}-6a^{5}z^{3}+8a^{3}z^{3}+32az^{3}+9z^{3}a^{-1}-7z^{3}a^{-3}-2a^{6}z^{2}+2a^{4}z^{2}+7a^{2}z^{2}-z^{2}a^{-2}-2z^{2}a^{-4}+4z^{2}+a^{5}z-2a^{3}z-6az-2za^{-1}+za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     2   2 6                   4 1   -3 4                 7 2     5 2               7 4       -3 0             10 7         3 -2           8 9           1 -4         7 8             -1 -6       4 8               4 -8     2 7                 -5 -10   1 4                   3 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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