L11a75

L11a74

L11a76

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Link Presentations

[edit Notes on L11a75's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(2)^{5}+3t(1)t(2)^{4}-4t(2)^{4}-4t(1)t(2)^{3}+4t(2)^{3}+4t(1)t(2)^{2}-4t(2)^{2}-4t(1)t(2)+3t(2)+2t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {2}{q^{25/2}}}+{\frac {5}{q^{23/2}}}-{\frac {7}{q^{21/2}}}+{\frac {9}{q^{19/2}}}-{\frac {11}{q^{17/2}}}+{\frac {10}{q^{15/2}}}-{\frac {10}{q^{13/2}}}+{\frac {6}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}-2a^{13}z^{-1}+3z^{3}a^{11}+8za^{11}+4a^{11}z^{-1}-2z^{5}a^{9}-6z^{3}a^{9}-4za^{9}-a^{9}z^{-1}-2z^{5}a^{7}-6z^{3}a^{7}-4za^{7}-a^{7}z^{-1}-z^{5}a^{5}-3z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$-z^{6}a^{16}+4z^{4}a^{16}-5z^{2}a^{16}+2a^{16}-2z^{7}a^{15}+6z^{5}a^{15}-3z^{3}a^{15}-za^{15}-2z^{8}a^{14}+3z^{6}a^{14}+5z^{4}a^{14}-5z^{2}a^{14}+a^{14}-2z^{9}a^{13}+5z^{7}a^{13}-7z^{5}a^{13}+13z^{3}a^{13}-8za^{13}+2a^{13}z^{-1}-z^{10}a^{12}+7z^{6}a^{12}-17z^{4}a^{12}+19z^{2}a^{12}-6a^{12}-5z^{9}a^{11}+21z^{7}a^{11}-40z^{5}a^{11}+34z^{3}a^{11}-15za^{11}+4a^{11}z^{-1}-z^{10}a^{10}-z^{8}a^{10}+13z^{6}a^{10}-29z^{4}a^{10}+18z^{2}a^{10}-5a^{10}-3z^{9}a^{9}+11z^{7}a^{9}-17z^{5}a^{9}+6z^{3}a^{9}-2za^{9}+a^{9}z^{-1}-3z^{8}a^{8}+8z^{6}a^{8}-6z^{4}a^{8}-2z^{2}a^{8}+a^{8}-3z^{7}a^{7}+9z^{5}a^{7}-9z^{3}a^{7}+5za^{7}-a^{7}z^{-1}-2z^{6}a^{6}+5z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+3z^{3}a^{5}-za^{5}$ (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                     2 1 -1 -8                   2     2 -10                 4 2     -2 -12               6 2       4 -14             5 5         0 -16           6 5           1 -18         3 5             2 -20       4 6               -2 -22     1 3                 2 -24   1 4                   -3 -26   1                     1 -28 1                       -1

Integral Khovanov Homology (db, data source)

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

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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