L11a6

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Link Presentations

[edit Notes on L11a6's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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