L11a70

L11a69

L11a71

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Link Presentations

[edit Notes on L11a70's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{5}-2t(2)^{5}-5t(1)t(2)^{4}+7t(2)^{4}+10t(1)t(2)^{3}-10t(2)^{3}-10t(1)t(2)^{2}+10t(2)^{2}+7t(1)t(2)-5t(2)-2t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {22}{q^{9/2}}}-{\frac {23}{q^{7/2}}}+{\frac {20}{q^{5/2}}}+q^{3/2}-{\frac {16}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {13}{q^{13/2}}}-{\frac {19}{q^{11/2}}}-5{\sqrt {q}}+{\frac {10}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}-3z^{3}a^{7}-6za^{7}-3a^{7}z^{-1}+3z^{5}a^{5}+8z^{3}a^{5}+8za^{5}+4a^{5}z^{-1}-z^{7}a^{3}-3z^{5}a^{3}-4z^{3}a^{3}-4za^{3}-2a^{3}z^{-1}+z^{5}a+z^{3}a-za$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-5a^{10}z^{4}+3a^{10}z^{2}-a^{10}+5a^{9}z^{7}-6a^{9}z^{5}+3a^{9}z^{3}-2a^{9}z+a^{9}z^{-1}+6a^{8}z^{8}-4a^{8}z^{6}-4a^{8}z^{4}+8a^{8}z^{2}-3a^{8}+5a^{7}z^{9}+2a^{7}z^{7}-17a^{7}z^{5}+24a^{7}z^{3}-13a^{7}z+3a^{7}z^{-1}+2a^{6}z^{10}+12a^{6}z^{8}-28a^{6}z^{6}+16a^{6}z^{4}+4a^{6}z^{2}-3a^{6}+12a^{5}z^{9}-14a^{5}z^{7}-15a^{5}z^{5}+30a^{5}z^{3}-18a^{5}z+4a^{5}z^{-1}+2a^{4}z^{10}+15a^{4}z^{8}-42a^{4}z^{6}+26a^{4}z^{4}-2a^{4}z^{2}-2a^{4}+7a^{3}z^{9}-6a^{3}z^{7}-15a^{3}z^{5}+15a^{3}z^{3}-7a^{3}z+2a^{3}z^{-1}+9a^{2}z^{8}-20a^{2}z^{6}+10a^{2}z^{4}-a^{2}z^{2}+5az^{7}-10az^{5}+4az^{3}+az+z^{6}-z^{4}$ (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                     4   4 0                   6 1   -5 -2                 10 4     6 -4               11 7       -4 -6             12 9         3 -8           10 11           1 -10         9 12             -3 -12       5 11               6 -14     2 8                 -6 -16   1 5                   4 -18   2                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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