L11a73

L11a72

L11a74

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Link Presentations

[edit Notes on L11a73's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(2)^{5}+4t(1)t(2)^{4}-5t(2)^{4}-7t(1)t(2)^{3}+7t(2)^{3}+7t(1)t(2)^{2}-7t(2)^{2}-5t(1)t(2)+4t(2)+2t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {10}{q^{11/2}}}-{\frac {15}{q^{13/2}}}+{\frac {16}{q^{15/2}}}-{\frac {16}{q^{17/2}}}+{\frac {13}{q^{19/2}}}-{\frac {10}{q^{21/2}}}+{\frac {6}{q^{23/2}}}-{\frac {2}{q^{25/2}}}+{\frac {1}{q^{27/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-2a^{13}z^{-1}+3a^{11}z^{3}+7a^{11}z+4a^{11}z^{-1}-2a^{9}z^{5}-4a^{9}z^{3}-a^{9}z-a^{9}z^{-1}-3a^{7}z^{5}-9a^{7}z^{3}-7a^{7}z-a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}$ (db)
Kauffman polynomial	$a^{16}z^{6}-4a^{16}z^{4}+5a^{16}z^{2}-2a^{16}+2a^{15}z^{7}-5a^{15}z^{5}+2a^{15}z^{3}+a^{15}z+3a^{14}z^{8}-6a^{14}z^{6}+a^{14}z^{4}+2a^{14}z^{2}-a^{14}+3a^{13}z^{9}-6a^{13}z^{7}+9a^{13}z^{5}-16a^{13}z^{3}+9a^{13}z-2a^{13}z^{-1}+a^{12}z^{10}+6a^{12}z^{8}-22a^{12}z^{6}+32a^{12}z^{4}-25a^{12}z^{2}+6a^{12}+7a^{11}z^{9}-18a^{11}z^{7}+29a^{11}z^{5}-31a^{11}z^{3}+17a^{11}z-4a^{11}z^{-1}+a^{10}z^{10}+9a^{10}z^{8}-30a^{10}z^{6}+41a^{10}z^{4}-22a^{10}z^{2}+5a^{10}+4a^{9}z^{9}-4a^{9}z^{7}-a^{9}z^{5}+4a^{9}z^{3}+2a^{9}z-a^{9}z^{-1}+6a^{8}z^{8}-12a^{8}z^{6}+9a^{8}z^{4}-a^{8}+6a^{7}z^{7}-15a^{7}z^{5}+15a^{7}z^{3}-7a^{7}z+a^{7}z^{-1}+3a^{6}z^{6}-5a^{6}z^{4}+a^{5}z^{5}-2a^{5}z^{3}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     3 1 -2 -8                   4     4 -10                 6 3     -3 -12               9 4       5 -14             8 7         -1 -16           8 8           0 -18         5 8             3 -20       5 8               -3 -22     1 5                 4 -24   1 5                   -4 -26   1                     1 -28 1                       -1

Integral Khovanov Homology (db, data source)

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

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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