L10n93

L10n92

L10n94

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Link Presentations

[edit Notes on L10n93's Link Presentations]

Planar diagram presentation	X8192 X9,18,10,19 X17,6,18,1 X16,7,17,8 X3,10,4,11 X5,14,6,15 X13,4,14,5 X20,11,13,12 X12,15,7,16 X2,19,3,20
Gauss code	{1, -10, -5, 7, -6, 3}, {4, -1, -2, 5, 8, -9}, {-7, 6, 9, -4, -3, 2, 10, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(uvw+1)\left(uvw^{2}-1\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{-11}+q^{-9}+q^{-6}+q^{-4}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$z^{2}a^{12}+a^{12}z^{-2}+2a^{12}-z^{6}a^{10}-7z^{4}a^{10}-14z^{2}a^{10}-2a^{10}z^{-2}-9a^{10}+z^{8}a^{8}+8z^{6}a^{8}+21z^{4}a^{8}+21z^{2}a^{8}+a^{8}z^{-2}+7a^{8}$ (db)
Kauffman polynomial	$a^{14}z^{2}-2a^{14}-a^{12}z^{2}-a^{12}z^{-2}+3a^{12}+a^{11}z^{7}-7a^{11}z^{5}+14a^{11}z^{3}-9a^{11}z+2a^{11}z^{-1}+a^{10}z^{8}-8a^{10}z^{6}+21a^{10}z^{4}-23a^{10}z^{2}-2a^{10}z^{-2}+11a^{10}+a^{9}z^{7}-7a^{9}z^{5}+14a^{9}z^{3}-9a^{9}z+2a^{9}z^{-1}+a^{8}z^{8}-8a^{8}z^{6}+21a^{8}z^{4}-21a^{8}z^{2}-a^{8}z^{-2}+7a^{8}$ (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 -7                 1 1 -9                 1 1 -11             1     1 -13         1         1 -15         1 1       0 -17     2 1           1 -19     2 1           1 -21 1                 1 -23 1                 1

Integral Khovanov Homology (db, data source)

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

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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