L11a313

L11a312

L11a314

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Link Presentations

[edit Notes on L11a313's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,6,19,5 X14,8,15,7 X20,13,21,14 X8,21,1,22 X16,10,17,9 X22,16,9,15 X6,18,7,17 X4,20,5,19
Gauss code	{1, -2, 3, -11, 4, -10, 5, -7}, {8, -1, 2, -3, 6, -5, 9, -8, 10, -4, 11, -6, 7, -9}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)(t(1)+t(2))(t(1)t(2)+1)\left(t(2)^{2}-t(2)+1\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{11/2}+3q^{9/2}-6q^{7/2}+10q^{5/2}-13q^{3/2}+14{\sqrt {q}}-{\frac {16}{\sqrt {q}}}+{\frac {13}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-z^{5}a^{-3}-3a^{3}z^{3}-3z^{3}a^{-3}-2a^{3}z-2za^{-3}+az^{7}+z^{7}a^{-1}+4az^{5}+4z^{5}a^{-1}+5az^{3}+5z^{3}a^{-1}+2az+2za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-5az^{9}-9z^{9}a^{-1}-4z^{9}a^{-3}-7a^{2}z^{8}+z^{8}a^{-2}-3z^{8}a^{-4}-3z^{8}-7a^{3}z^{7}+7az^{7}+30z^{7}a^{-1}+15z^{7}a^{-3}-z^{7}a^{-5}-5a^{4}z^{6}+13a^{2}z^{6}+13z^{6}a^{-2}+12z^{6}a^{-4}+19z^{6}-3a^{5}z^{5}+12a^{3}z^{5}+4az^{5}-33z^{5}a^{-1}-18z^{5}a^{-3}+4z^{5}a^{-5}-a^{6}z^{4}+5a^{4}z^{4}-9a^{2}z^{4}-16z^{4}a^{-2}-14z^{4}a^{-4}-17z^{4}+3a^{5}z^{3}-11a^{3}z^{3}-11az^{3}+19z^{3}a^{-1}+12z^{3}a^{-3}-4z^{3}a^{-5}+a^{6}z^{2}-a^{4}z^{2}+7z^{2}a^{-2}+5z^{2}a^{-4}+4z^{2}+4a^{3}z+4az-4za^{-1}-4za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     2   -2 8                   4 1   3 6                 6 2     -4 4               7 4       3 2             7 6         -1 0           9 7           2 -2         6 9             3 -4       4 7               -3 -6     2 6                 4 -8   1 4                   -3 -10   2                     2 -12 1                       -1

Integral Khovanov Homology (db, data source)

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

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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