L11n95

L11n94

L11n96

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Link Presentations

[edit Notes on L11n95's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,16,8,17 X9,18,10,19 X17,8,18,9 X19,22,20,5 X13,20,14,21 X21,14,22,15 X15,10,16,11 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 5, -4, 9, 11, -2, -7, 8, -9, 3, -5, 4, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{5}+t(2)^{5}-2t(2)^{4}-t(1)t(2)^{3}+2t(2)^{3}+2t(1)t(2)^{2}-t(2)^{2}-2t(1)t(2)+t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{27/2}}}-{\frac {2}{q^{25/2}}}+{\frac {3}{q^{23/2}}}-{\frac {3}{q^{21/2}}}+{\frac {4}{q^{19/2}}}-{\frac {4}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {2}{q^{13/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}-a^{13}z^{-1}+2z^{3}a^{11}+4za^{11}+a^{11}z^{-1}+2z^{3}a^{9}+4za^{9}+2a^{9}z^{-1}-z^{7}a^{7}-7z^{5}a^{7}-14z^{3}a^{7}-9za^{7}-2a^{7}z^{-1}$ (db)
Kauffman polynomial	$a^{16}z^{6}-4a^{16}z^{4}+3a^{16}z^{2}-a^{16}+2a^{15}z^{7}-8a^{15}z^{5}+6a^{15}z^{3}-a^{15}z+2a^{14}z^{8}-8a^{14}z^{6}+6a^{14}z^{4}+a^{13}z^{9}-3a^{13}z^{7}-a^{13}z^{5}+3a^{13}z^{3}+a^{13}z-a^{13}z^{-1}+3a^{12}z^{8}-16a^{12}z^{6}+26a^{12}z^{4}-15a^{12}z^{2}+3a^{12}+a^{11}z^{9}-6a^{11}z^{7}+12a^{11}z^{5}-10a^{11}z^{3}+4a^{11}z-a^{11}z^{-1}+a^{10}z^{8}-7a^{10}z^{6}+14a^{10}z^{4}-7a^{10}z^{2}-2a^{9}z^{5}+7a^{9}z^{3}-7a^{9}z+2a^{9}z^{-1}-2a^{8}z^{4}+5a^{8}z^{2}-3a^{8}+a^{7}z^{7}-7a^{7}z^{5}+14a^{7}z^{3}-9a^{7}z+2a^{7}z^{-1}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                       1 1 -8                       1 1 -10                 1 1     0 -12               2         2 -14             2 3 1       0 -16           3 1           2 -18         2 3 1           0 -20       2 3               -1 -22     1 2 1               0 -24   1 2                   -1 -26   1                     1 -28 1                       -1

Integral Khovanov Homology (db, data source)

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

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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