L11n62

L11n61

L11n63

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Link Presentations

[edit Notes on L11n62's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X15,20,16,21 X11,18,12,19 X19,12,20,13 X17,22,18,5 X21,16,22,17 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 6, -9, 3, -4, 8, -7, 5, -6, 4, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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