L10a95

L10a94

L10a96

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Link Presentations

[edit Notes on L10a95's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(1)+t(2)-1)(t(2)t(1)-t(1)-t(2))}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {5}{q^{9/2}}}-q^{7/2}+{\frac {9}{q^{7/2}}}+3q^{5/2}-{\frac {12}{q^{5/2}}}-6q^{3/2}+{\frac {11}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {3}{q^{11/2}}}+9{\sqrt {q}}-{\frac {12}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z-a^{3}z^{5}-2a^{3}z^{3}-3a^{3}z-za^{-3}-az^{5}+2z^{3}a^{-1}+2az+az^{-1}+za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+3a^{6}z^{6}-7a^{6}z^{4}+4a^{6}z^{2}+4a^{5}z^{7}-8a^{5}z^{5}+6a^{5}z^{3}-3a^{5}z+3a^{4}z^{8}-a^{4}z^{6}-7a^{4}z^{4}+8a^{4}z^{2}+a^{3}z^{9}+7a^{3}z^{7}-19a^{3}z^{5}+z^{5}a^{-3}+21a^{3}z^{3}-2z^{3}a^{-3}-10a^{3}z+za^{-3}+6a^{2}z^{8}-7a^{2}z^{6}+3z^{6}a^{-2}-a^{2}z^{4}-6z^{4}a^{-2}+5a^{2}z^{2}+3z^{2}a^{-2}+az^{9}+7az^{7}+4z^{7}a^{-1}-17az^{5}-6z^{5}a^{-1}+17az^{3}+2z^{3}a^{-1}-10az-2za^{-1}+az^{-1}+a^{-1}z^{-1}+3z^{8}-7z^{4}+4z^{2}-1$ (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                   2   -2 4                 4 1   3 2               5 2     -3 0             7 4       3 -2           6 7         1 -4         6 5           1 -6       3 6             3 -8     2 6               -4 -10   1 3                 2 -12   2                   -2 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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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