L11a206

L11a205

L11a207

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Link Presentations

[edit Notes on L11a206's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X18,15,19,16 X16,6,17,5 X4,18,5,17 X22,11,7,12 X20,13,21,14 X14,19,15,20 X12,21,13,22
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 8, -11, 9, -10, 5, -6, 7, -5, 10, -9, 11, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {3t(2)^{2}t(1)^{2}-3t(2)t(1)^{2}-3t(2)^{2}t(1)+5t(2)t(1)-3t(1)-3t(2)+3}{t(1)t(2)}}$ (db)
Jones polynomial	${\frac {7}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {4}{q^{5/2}}}-{\frac {3}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {5}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {7}{q^{11/2}}}-{\sqrt {q}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{9}-2za^{9}+z^{5}a^{7}+3z^{3}a^{7}+2za^{7}+z^{5}a^{5}+2z^{3}a^{5}+z^{5}a^{3}+3z^{3}a^{3}+2za^{3}+a^{3}z^{-1}-z^{3}a-3za-az^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+4z^{4}a^{12}-3z^{2}a^{12}-2z^{7}a^{11}+8z^{5}a^{11}-7z^{3}a^{11}+za^{11}-2z^{8}a^{10}+7z^{6}a^{10}-5z^{4}a^{10}+z^{2}a^{10}-2z^{9}a^{9}+9z^{7}a^{9}-15z^{5}a^{9}+12z^{3}a^{9}-2za^{9}-z^{10}a^{8}+4z^{8}a^{8}-7z^{6}a^{8}+6z^{4}a^{8}-2z^{2}a^{8}-3z^{9}a^{7}+15z^{7}a^{7}-29z^{5}a^{7}+19z^{3}a^{7}-3za^{7}-z^{10}a^{6}+5z^{8}a^{6}-12z^{6}a^{6}+11z^{4}a^{6}-6z^{2}a^{6}-z^{9}a^{5}+3z^{7}a^{5}-4z^{5}a^{5}-z^{3}a^{5}+za^{5}-z^{8}a^{4}+2z^{6}a^{4}-2z^{4}a^{4}+z^{2}a^{4}-z^{7}a^{3}+z^{5}a^{3}+3z^{3}a^{3}-3za^{3}+a^{3}z^{-1}-z^{6}a^{2}+2z^{4}a^{2}+z^{2}a^{2}-a^{2}-z^{5}a+4z^{3}a-4za+az^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                         0 -2                   3 1   2 -4                 2 1     -1 -6               4 2       2 -8             4 3         -1 -10           3 3           0 -12         3 4             1 -14       2 3               -1 -16     1 3                 2 -18   1 2                   -1 -20   1                     1 -22 1                       -1

Integral Khovanov Homology (db, data source)

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

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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