L11a148

L11a147

L11a149

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Link Presentations

[edit Notes on L11a148's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X20,11,21,12 X12,6,13,5 X4,20,5,19 X14,18,15,17 X16,8,17,7 X22,16,7,15 X18,14,19,13 X6,21,1,22
Gauss code	{1, -2, 3, -6, 5, -11}, {8, -1, 2, -3, 4, -5, 10, -7, 9, -8, 7, -10, 6, -4, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{4}-5u^{2}v^{3}+5u^{2}v^{2}-2u^{2}v-2uv^{4}+6uv^{3}-9uv^{2}+6uv-2u-2v^{3}+5v^{2}-5v+2}{uv^{2}}}$ (db)
Jones polynomial	$4q^{9/2}-{\frac {3}{q^{9/2}}}-7q^{7/2}+{\frac {6}{q^{7/2}}}+11q^{5/2}-{\frac {11}{q^{5/2}}}-15q^{3/2}+{\frac {14}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+16{\sqrt {q}}-{\frac {17}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-z^{5}a^{-3}-3a^{3}z^{3}-2z^{3}a^{-3}-2a^{3}z+za^{-3}+a^{-3}z^{-1}+az^{7}+z^{7}a^{-1}+4az^{5}+3z^{5}a^{-1}+6az^{3}+5az-5za^{-1}+2az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-a^{6}z^{2}+z^{7}a^{-5}+3a^{5}z^{5}-3z^{5}a^{-5}-3a^{5}z^{3}+2z^{3}a^{-5}+a^{5}z+4z^{8}a^{-4}+5a^{4}z^{6}-15z^{6}a^{-4}-4a^{4}z^{4}+15z^{4}a^{-4}+a^{4}z^{2}-2z^{2}a^{-4}-a^{-4}+5z^{9}a^{-3}+7a^{3}z^{7}-17z^{7}a^{-3}-9a^{3}z^{5}+15z^{5}a^{-3}+6a^{3}z^{3}-3z^{3}a^{-3}-a^{3}z-za^{-3}+a^{-3}z^{-1}+2z^{10}a^{-2}+7a^{2}z^{8}+3z^{8}a^{-2}-9a^{2}z^{6}-26z^{6}a^{-2}+2a^{2}z^{4}+25z^{4}a^{-2}+2a^{2}z^{2}-2z^{2}a^{-2}-3a^{-2}+5az^{9}+10z^{9}a^{-1}-3az^{7}-28z^{7}a^{-1}-12az^{5}+18z^{5}a^{-1}+15az^{3}+z^{3}a^{-1}-8az-7za^{-1}+2az^{-1}+3a^{-1}z^{-1}+2z^{10}+6z^{8}-25z^{6}+17z^{4}-3$ (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 12                       1 1 10                     3   -3 8                   4 1   3 6                 7 3     -4 4               8 4       4 2             8 7         -1 0           9 8           1 -2         6 9             3 -4       5 8               -3 -6     2 7                 5 -8   1 4                   -3 -10   2                     2 -12 1                       -1

Integral Khovanov Homology (db, data source)

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

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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