L11a45

L11a44

L11a46

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Link Presentations

[edit Notes on L11a45's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)^{2}}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+15{\sqrt {q}}-{\frac {21}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {24}{q^{5/2}}}+{\frac {20}{q^{7/2}}}-{\frac {14}{q^{9/2}}}+{\frac {9}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+3a^{3}z^{5}+3az^{5}-z^{5}a^{-1}-2a^{5}z^{3}+3a^{3}z^{3}+3az^{3}-2z^{3}a^{-1}-a^{5}z+2a^{3}z-za^{-1}-a^{5}z^{-1}+3a^{3}z^{-1}-2az^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-6a^{5}z^{9}-12a^{3}z^{9}-6az^{9}-7a^{6}z^{8}-12a^{4}z^{8}-14a^{2}z^{8}-9z^{8}-4a^{7}z^{7}+7a^{5}z^{7}+15a^{3}z^{7}-4az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+16a^{6}z^{6}+31a^{4}z^{6}+27a^{2}z^{6}-4z^{6}a^{-2}+9z^{6}+9a^{7}z^{5}+2a^{5}z^{5}+20az^{5}+12z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-11a^{6}z^{4}-19a^{4}z^{4}-10a^{2}z^{4}+5z^{4}a^{-2}+z^{4}-5a^{7}z^{3}+a^{3}z^{3}-13az^{3}-8z^{3}a^{-1}+z^{3}a^{-3}-a^{8}z^{2}+2a^{6}z^{2}+5a^{4}z^{2}+a^{2}z^{2}-2z^{2}a^{-2}-3z^{2}-2a^{5}z-3a^{3}z+az+2za^{-1}-a^{6}-3a^{4}-3a^{2}+a^{5}z^{-1}+3a^{3}z^{-1}+2az^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 9 3     -6 0               12 6       6 -2             12 11         -1 -4           12 10           2 -6         8 12             4 -8       6 12               -6 -10     3 8                 5 -12   1 6                   -5 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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