L11n29

L11n28

L11n30

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Link Presentations

[edit Notes on L11n29's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X5,12,6,13 X3849 X9,16,10,17 X13,22,14,5 X15,10,16,11 X21,14,22,15 X11,20,12,21 X17,2,18,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 8, -10, 4, -7, 9, -8, 6, -11, -2, 3, 10, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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