L11n94

L11n93

L11n95

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Link Presentations

[edit Notes on L11n94's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X9,14,10,15 X19,22,20,5 X11,21,12,20 X21,11,22,10 X15,19,16,18 X7,17,8,16 X17,9,18,8 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -8, 9, -3, 6, -5, -2, 11, 3, -7, 8, -9, 7, -4, 5, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(v^{2}+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+2q^{5/2}-{\frac {1}{q^{5/2}}}-4q^{3/2}+{\frac {2}{q^{3/2}}}+q^{15/2}-q^{13/2}+3{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$za^{-7}+a^{-7}z^{-1}-z^{3}a^{-5}-4za^{-5}-3a^{-5}z^{-1}+2z^{3}a^{-3}+6za^{-3}+3a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-4z^{3}a^{-1}+2az-5za^{-1}+az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-4z^{8}a^{-2}-2z^{8}a^{-4}-2z^{8}-az^{7}+2z^{7}a^{-1}+3z^{7}a^{-3}-z^{7}a^{-5}-z^{7}a^{-7}+21z^{6}a^{-2}+12z^{6}a^{-4}-z^{6}a^{-8}+10z^{6}+5az^{5}+9z^{5}a^{-1}+6z^{5}a^{-3}+8z^{5}a^{-5}+6z^{5}a^{-7}-29z^{4}a^{-2}-18z^{4}a^{-4}+3z^{4}a^{-6}+5z^{4}a^{-8}-13z^{4}-7az^{3}-19z^{3}a^{-1}-18z^{3}a^{-3}-14z^{3}a^{-5}-8z^{3}a^{-7}+13z^{2}a^{-2}+8z^{2}a^{-4}-5z^{2}a^{-6}-5z^{2}a^{-8}+5z^{2}+4az+11za^{-1}+14za^{-3}+10za^{-5}+3za^{-7}-2a^{-2}+2a^{-6}+a^{-8}-az^{-1}-2a^{-1}z^{-1}-3a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                         0 12                 1 1 1   1 10               2 1       -1 8             2 2 1       1 6           2 3 1         0 4         3 2 1           2 2       2 4 1             1 0     1 2 1               0 -2   1 2                   1 -4   1                     -1 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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