L10a96

L10a95

L10a97

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Link Presentations

[edit Notes on L10a96's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X14,5,15,6 X16,19,17,20 X18,7,19,8 X6,17,7,18 X20,15,9,16 X8,13,1,14
Gauss code	{1, -2, 3, -4, 5, -8, 7, -10}, {4, -1, 2, -3, 10, -5, 9, -6, 8, -7, 6, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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