L11n64

L11n63

L11n65

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Link Presentations

[edit Notes on L11n64's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X15,20,16,21 X11,18,12,19 X19,12,20,13 X17,22,18,5 X21,16,22,17 X8,14,9,13 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	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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