L11a50

L11a49

L11a51

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Link Presentations

[edit Notes on L11a50's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X12,6,13,5 X10,4,11,3 X22,14,5,13 X14,22,15,21 X20,12,21,11 X16,9,17,10 X2,16,3,15 X8,17,9,18
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 8, -4, 6, -7, 10, -9, 11, -2, 3, -8, 7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2(u-1)(v-1)\left(2v^{2}-3v+2\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-7q^{9/2}+{\frac {1}{q^{9/2}}}+11q^{7/2}-{\frac {4}{q^{7/2}}}-15q^{5/2}+{\frac {7}{q^{5/2}}}+18q^{3/2}-{\frac {12}{q^{3/2}}}-q^{13/2}+3q^{11/2}-18{\sqrt {q}}+{\frac {15}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-za^{-5}-a^{-5}z^{-1}+z^{5}a^{-3}-a^{3}z^{3}+z^{3}a^{-3}+2za^{-3}+a^{3}z^{-1}+2a^{-3}z^{-1}+az^{5}+2z^{5}a^{-1}+3z^{3}a^{-1}-2az+za^{-1}-az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-2z^{3}a^{-7}+3z^{6}a^{-6}-5z^{4}a^{-6}+6z^{7}a^{-5}-14z^{5}a^{-5}+12z^{3}a^{-5}-5za^{-5}+a^{-5}z^{-1}+7z^{8}a^{-4}+a^{4}z^{6}-17z^{6}a^{-4}-2a^{4}z^{4}+19z^{4}a^{-4}-8z^{2}a^{-4}+a^{-4}+5z^{9}a^{-3}+4a^{3}z^{7}-7z^{7}a^{-3}-11a^{3}z^{5}-z^{5}a^{-3}+6a^{3}z^{3}+15z^{3}a^{-3}+a^{3}z-11za^{-3}-a^{3}z^{-1}+2a^{-3}z^{-1}+2z^{10}a^{-2}+6a^{2}z^{8}+5z^{8}a^{-2}-16a^{2}z^{6}-22z^{6}a^{-2}+10a^{2}z^{4}+31z^{4}a^{-2}-2a^{2}z^{2}-15z^{2}a^{-2}+a^{2}+3a^{-2}+5az^{9}+10z^{9}a^{-1}-10az^{7}-27z^{7}a^{-1}+4az^{5}+29z^{5}a^{-1}-3az^{3}-8z^{3}a^{-1}+3az-4za^{-1}-az^{-1}+a^{-1}z^{-1}+2z^{10}+4z^{8}-19z^{6}+19z^{4}-9z^{2}+2$ (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                     2   -2 10                   5 1   4 8                 6 2     -4 6               9 5       4 4             9 6         -3 2           9 9           0 0         8 11             3 -2       4 7               -3 -4     3 8                 5 -6   1 4                   -3 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

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

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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