L11a18

L11a17

L11a19

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Link Presentations

[edit Notes on L11a18's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X18,14,19,13 X16,9,17,10 X8,17,9,18 X22,20,5,19 X20,15,21,16 X14,21,15,22 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -2, 11, -3, 4, -9, 8, -5, 6, -4, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)^{5}}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+9q^{7/2}-15q^{5/2}+18q^{3/2}-21{\sqrt {q}}+{\frac {20}{\sqrt {q}}}-{\frac {17}{q^{3/2}}}+{\frac {12}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+2z^{3}a^{-3}-6a^{3}z+2za^{-3}-3a^{3}z^{-1}-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+9az^{3}-8z^{3}a^{-1}+10az-7za^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-9az^{9}-6z^{9}a^{-1}-3a^{4}z^{8}-12a^{2}z^{8}-13z^{8}a^{-2}-22z^{8}-a^{5}z^{7}+3a^{3}z^{7}+7az^{7}-11z^{7}a^{-1}-14z^{7}a^{-3}+11a^{4}z^{6}+47a^{2}z^{6}+18z^{6}a^{-2}-9z^{6}a^{-4}+63z^{6}+4a^{5}z^{5}+16a^{3}z^{5}+41az^{5}+54z^{5}a^{-1}+21z^{5}a^{-3}-4z^{5}a^{-5}-14a^{4}z^{4}-51a^{2}z^{4}-5z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}-50z^{4}-6a^{5}z^{3}-31a^{3}z^{3}-61az^{3}-51z^{3}a^{-1}-14z^{3}a^{-3}+z^{3}a^{-5}+7a^{4}z^{2}+20a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+16z^{2}+4a^{5}z+18a^{3}z+28az+19za^{-1}+5za^{-3}-a^{4}-3a^{2}-a^{-2}-2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   6 1   -5 6                 9 3     6 4               9 6       -3 2             12 9         3 0           10 11           1 -2         7 10             -3 -4       5 10               5 -6     2 7                 -5 -8   1 5                   4 -10   2                     -2 -12 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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