L11a119

L11a118

L11a120

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Link Presentations

[edit Notes on L11a119's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X18,8,19,7 X22,20,5,19 X20,11,21,12 X10,21,11,22 X16,10,17,9 X12,16,13,15 X8,18,9,17 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 7, -6, 5, -8, 11, -2, 8, -7, 9, -3, 4, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(v-2)(2v-1)(uv-2u-2v+1)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+7q^{5/2}-12q^{3/2}+16{\sqrt {q}}-{\frac {17}{\sqrt {q}}}+{\frac {17}{q^{3/2}}}-{\frac {15}{q^{5/2}}}+{\frac {10}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-3za^{5}-a^{5}z^{-1}+3z^{3}a^{3}-a^{3}z^{-1}-z^{5}a+2z^{3}a+4za+2az^{-1}-z^{5}a^{-1}-z^{3}a^{-1}-3za^{-1}-a^{-1}z^{-1}+z^{3}a^{-3}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-7az^{9}-4z^{9}a^{-1}-4a^{4}z^{8}-9a^{2}z^{8}-6z^{8}a^{-2}-11z^{8}-3a^{5}z^{7}-a^{3}z^{7}+8az^{7}+2z^{7}a^{-1}-4z^{7}a^{-3}-2a^{6}z^{6}+5a^{4}z^{6}+25a^{2}z^{6}+16z^{6}a^{-2}-z^{6}a^{-4}+35z^{6}-a^{7}z^{5}+2a^{5}z^{5}+8a^{3}z^{5}+10az^{5}+16z^{5}a^{-1}+11z^{5}a^{-3}+3a^{6}z^{4}-6a^{4}z^{4}-30a^{2}z^{4}-12z^{4}a^{-2}+2z^{4}a^{-4}-35z^{4}+3a^{7}z^{3}+3a^{5}z^{3}-12a^{3}z^{3}-23az^{3}-18z^{3}a^{-1}-7z^{3}a^{-3}+5a^{4}z^{2}+16a^{2}z^{2}+4z^{2}a^{-2}+15z^{2}-3a^{7}z-3a^{5}z+7a^{3}z+13az+6za^{-1}-a^{6}-2a^{4}-3a^{2}-1+a^{7}z^{-1}+a^{5}z^{-1}-a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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                     3   3 6                   4 1   -3 4                 8 3     5 2               8 4       -4 0             9 8         1 -2           9 9           0 -4         6 8             -2 -6       4 9               5 -8     2 6                 -4 -10   1 5                   4 -12   1                     -1 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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