L11a190

L11a189

L11a191

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Link Presentations

[edit Notes on L11a190's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X18,13,19,14 X22,20,7,19 X20,12,21,11 X12,22,13,21 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 8, -9, 6, -3, 5, -4, 3, -6, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)t(2)^{4}-t(2)^{4}+2t(1)^{2}t(2)^{3}-6t(1)t(2)^{3}+4t(2)^{3}-4t(1)^{2}t(2)^{2}+9t(1)t(2)^{2}-4t(2)^{2}+4t(1)^{2}t(2)-6t(1)t(2)+2t(2)-t(1)^{2}+2t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{5/2}-3q^{3/2}+6{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {13}{q^{3/2}}}-{\frac {15}{q^{5/2}}}+{\frac {14}{q^{7/2}}}-{\frac {13}{q^{9/2}}}+{\frac {9}{q^{11/2}}}-{\frac {6}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{7}+za^{7}-z^{5}a^{5}-z^{3}a^{5}+za^{5}+2a^{5}z^{-1}-2z^{5}a^{3}-5z^{3}a^{3}-6za^{3}-3a^{3}z^{-1}-z^{5}a-z^{3}a+az^{-1}+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-4a^{9}z^{5}+5a^{9}z^{3}-2a^{9}z+3a^{8}z^{8}-12a^{8}z^{6}+14a^{8}z^{4}-4a^{8}z^{2}+3a^{7}z^{9}-8a^{7}z^{7}+a^{7}z^{5}+6a^{7}z^{3}-a^{7}z+a^{6}z^{10}+5a^{6}z^{8}-24a^{6}z^{6}+23a^{6}z^{4}-6a^{6}z^{2}+6a^{5}z^{9}-11a^{5}z^{7}-6a^{5}z^{5}+15a^{5}z^{3}-9a^{5}z+2a^{5}z^{-1}+a^{4}z^{10}+7a^{4}z^{8}-18a^{4}z^{6}+4a^{4}z^{4}+6a^{4}z^{2}-3a^{4}+3a^{3}z^{9}+4a^{3}z^{7}-20a^{3}z^{5}+20a^{3}z^{3}-12a^{3}z+3a^{3}z^{-1}+5a^{2}z^{8}-a^{2}z^{6}-11a^{2}z^{4}+z^{4}a^{-2}+12a^{2}z^{2}-z^{2}a^{-2}-3a^{2}+6az^{7}-6az^{5}+3z^{5}a^{-1}+3az^{3}-3z^{3}a^{-1}-az+za^{-1}+az^{-1}+5z^{6}-5z^{4}+3z^{2}-1$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                       1 -1 4                     2   2 2                   4 1   -3 0                 6 2     4 -2               8 5       -3 -4             7 5         2 -6           7 8           1 -8         6 7             -1 -10       3 7               4 -12     3 6                 -3 -14   1 4                   3 -16   2                     -2 -18 1                       1

Integral Khovanov Homology (db, data source)

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

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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