L11n26

L11n25

L11n27

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Link Presentations

[edit Notes on L11n26's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X13,22,14,5 X21,14,22,15 X11,18,12,19 X9,20,10,21 X19,10,20,11 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -9, 10, -8, 4, -6, 7, -11, -2, 3, 8, -10, 9, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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