L11a262

L11a261

L11a263

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Link Presentations

[edit Notes on L11a262's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)t(1)^{2}-t(1)^{2}+2t(2)^{2}t(1)-4t(2)t(1)+2t(1)-t(2)^{2}+2t(2)\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-8q^{9/2}+{\frac {1}{q^{9/2}}}+12q^{7/2}-{\frac {3}{q^{7/2}}}-16q^{5/2}+{\frac {6}{q^{5/2}}}+18q^{3/2}-{\frac {11}{q^{3/2}}}-q^{13/2}+4q^{11/2}-18{\sqrt {q}}+{\frac {14}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}+z^{5}a^{-3}-a^{3}z^{3}-a^{3}z-za^{-3}+az^{5}+2z^{5}a^{-1}+az^{3}+3z^{3}a^{-1}+az+za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-6z^{4}a^{-6}+z^{2}a^{-6}+7z^{7}a^{-5}-12z^{5}a^{-5}+5z^{3}a^{-5}-za^{-5}+7z^{8}a^{-4}+a^{4}z^{6}-10z^{6}a^{-4}-3a^{4}z^{4}+4z^{4}a^{-4}+2a^{4}z^{2}-z^{2}a^{-4}+4z^{9}a^{-3}+3a^{3}z^{7}+2z^{7}a^{-3}-9a^{3}z^{5}-13z^{5}a^{-3}+8a^{3}z^{3}+10z^{3}a^{-3}-3a^{3}z-2za^{-3}+z^{10}a^{-2}+4a^{2}z^{8}+11z^{8}a^{-2}-9a^{2}z^{6}-27z^{6}a^{-2}+4a^{2}z^{4}+23z^{4}a^{-2}-5z^{2}a^{-2}+3az^{9}+7z^{9}a^{-1}-az^{7}-9z^{7}a^{-1}-11az^{5}-2z^{5}a^{-1}+13az^{3}+9z^{3}a^{-1}-6az-4za^{-1}+az^{-1}+a^{-1}z^{-1}+z^{10}+8z^{8}-23z^{6}+20z^{4}-5z^{2}-1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   5 1   4 8                 7 3     -4 6               9 5       4 4             9 7         -2 2           9 9           0 0         7 11             4 -2       4 7               -3 -4     2 7                 5 -6   1 4                   -3 -8   2                     2 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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