L11n47

L11n46

L11n48

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Link Presentations

[edit Notes on L11n47's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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