L11a269

L11a268

L11a270

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Link Presentations

[edit Notes on L11a269's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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