L10a65

L10a64

L10a66

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Link Presentations

[edit Notes on L10a65's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X4,20,5,19 X14,5,15,6 X18,11,19,12 X16,13,17,14 X12,17,13,18 X6,15,1,16
Gauss code	{1, -4, 2, -5, 6, -10}, {4, -1, 3, -2, 7, -9, 8, -6, 10, -8, 9, -7, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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