L10a85

L10a84

L10a86

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Link Presentations

[edit Notes on L10a85's Link Presentations]

Planar diagram presentation	X8192 X16,5,17,6 X18,10,19,9 X10,20,11,19 X14,18,15,17 X2,11,3,12 X12,3,13,4 X4758 X20,14,7,13 X6,15,1,16
Gauss code	{1, -6, 7, -8, 2, -10}, {8, -1, 3, -4, 6, -7, 9, -5, 10, -2, 5, -3, 4, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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