L11a487

L11a486

L11a488

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Link Presentations

[edit Notes on L11a487's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X22,10,19,9 X8493 X16,21,17,22 X18,11,5,12 X20,5,21,6 X10,17,11,18 X12,20,13,19 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {10, -8, 6, -4}, {8, -1, 2, -5, 4, -9, 7, -10, 11, -2, 3, -6, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)^{3}\left(t(3)^{2}-t(3)+1\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{3}-5q^{2}+12q-18+28q^{-1}-30q^{-2}+32q^{-3}-27q^{-4}+20q^{-5}-13q^{-6}+5q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{4}-a^{6}z^{2}-a^{6}z^{-2}-a^{6}+2a^{4}z^{6}+5a^{4}z^{4}+4a^{4}z^{2}+4a^{4}z^{-2}+4a^{4}-a^{2}z^{8}-4a^{2}z^{6}-6a^{2}z^{4}-4a^{2}z^{2}-5a^{2}z^{-2}-5a^{2}+z^{6}+2z^{4}+z^{2}+2z^{-2}+2$ (db)
Kauffman polynomial	$a^{9}z^{5}+5a^{8}z^{6}-2a^{8}z^{4}+13a^{7}z^{7}-16a^{7}z^{5}+7a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+18a^{6}z^{8}-27a^{6}z^{6}+14a^{6}z^{4}-4a^{6}z^{2}-a^{6}z^{-2}+a^{6}+13a^{5}z^{9}-4a^{5}z^{7}-23a^{5}z^{5}+18a^{5}z^{3}-10a^{5}z+5a^{5}z^{-1}+4a^{4}z^{10}+28a^{4}z^{8}-72a^{4}z^{6}+49a^{4}z^{4}-13a^{4}z^{2}-4a^{4}z^{-2}+4a^{4}+24a^{3}z^{9}-38a^{3}z^{7}+3a^{3}z^{5}+14a^{3}z^{3}-12a^{3}z+9a^{3}z^{-1}+4a^{2}z^{10}+21a^{2}z^{8}-65a^{2}z^{6}+z^{6}a^{-2}+51a^{2}z^{4}-z^{4}a^{-2}-13a^{2}z^{2}-5a^{2}z^{-2}+5a^{2}+11az^{9}-16az^{7}+5z^{7}a^{-1}+az^{5}-8z^{5}a^{-1}+5az^{3}+2z^{3}a^{-1}-5az+5az^{-1}+11z^{8}-24z^{6}+17z^{4}-4z^{2}-2z^{-2}+3$ (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 4 χ 7                       1 1 5                     4   -4 3                   8 1   7 1                 10 4     -6 -1               18 8       10 -3             16 14         -2 -5           16 14           2 -7         11 16             5 -9       9 16               -7 -11     4 11                 7 -13   1 9                   -8 -15   4                     4 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-3$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-2$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{16}$ ${\mathbb {Z} }^{16}$ $r=-1$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{16}$ ${\mathbb {Z} }^{16}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{18}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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