L11n199

L11n198

L11n200

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Link Presentations

[edit Notes on L11n199's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X5,14,6,15 X16,7,17,8 X20,15,21,16 X13,18,14,19 X21,6,22,7 X17,22,18,9 X4,19,5,20 X2,9,3,10 X8,11,1,12
Gauss code	{1, -10, 2, -9, -3, 7, 4, -11}, {10, -1, 11, -2, -6, 3, 5, -4, -8, 6, 9, -5, -7, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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