L11n68

L11n67

L11n69

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Link Presentations

[edit Notes on L11n68's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{5}-uv^{3}+uv^{2}-uv+u+v^{5}-v^{4}+v^{3}-v^{2}+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{13/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {2}{q^{21/2}}}+{\frac {2}{q^{23/2}}}-{\frac {1}{q^{25/2}}}+{\frac {1}{q^{27/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}-2a^{13}z^{-1}+2z^{3}a^{11}+6za^{11}+4a^{11}z^{-1}+z^{3}a^{9}+za^{9}-a^{9}z^{-1}-z^{7}a^{7}-7z^{5}a^{7}-14z^{3}a^{7}-8za^{7}-a^{7}z^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{16}+5z^{4}a^{16}-6z^{2}a^{16}+2a^{16}-z^{7}a^{15}+4z^{5}a^{15}-2z^{3}a^{15}-za^{15}-z^{8}a^{14}+4z^{6}a^{14}-2z^{4}a^{14}-2z^{2}a^{14}+a^{14}-z^{9}a^{13}+6z^{7}a^{13}-13z^{5}a^{13}+16z^{3}a^{13}-10za^{13}+2a^{13}z^{-1}-2z^{8}a^{12}+13z^{6}a^{12}-28z^{4}a^{12}+24z^{2}a^{12}-6a^{12}-z^{9}a^{11}+8z^{7}a^{11}-23z^{5}a^{11}+30z^{3}a^{11}-18za^{11}+4a^{11}z^{-1}-z^{8}a^{10}+8z^{6}a^{10}-20z^{4}a^{10}+18z^{2}a^{10}-5a^{10}+z^{5}a^{9}-2z^{3}a^{9}-za^{9}+a^{9}z^{-1}+z^{4}a^{8}-2z^{2}a^{8}+a^{8}-z^{7}a^{7}+7z^{5}a^{7}-14z^{3}a^{7}+8za^{7}-a^{7}z^{-1}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                       1 1 -8                       1 1 -10                 1 1     0 -12               1         1 -14             2 3 1       0 -16           2             2 -18         1 3 1           1 -20       2 2               0 -22       1 1               0 -24   1 2                   -1 -26                         0 -28 1                       -1

Integral Khovanov Homology (db, data source)

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

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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