L11n154

L11n153

L11n155

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Link Presentations

[edit Notes on L11n154's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X7,16,8,17 X13,20,14,21 X15,22,16,7 X19,1,20,6 X18,11,19,12 X5,12,6,13 X21,14,22,15 X4,18,5,17
Gauss code	{1, -2, 3, -11, -9, 7}, {-4, -1, 2, -3, 8, 9, -5, 10, -6, 4, 11, -8, -7, 5, -10, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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