L11a163

L11a162

L11a164

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Link Presentations

[edit Notes on L11a163's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,11,19,12 X22,19,7,20 X20,15,21,16 X16,21,17,22 X12,17,13,18 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 5, -9, 11, -4, 7, -8, 9, -5, 6, -7, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-5t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-t(2)^{3}+4t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+4t(2)^{2}-t(1)^{2}t(2)+5t(1)t(2)-5t(2)-t(1)+3}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{29/2}}}-{\frac {3}{q^{27/2}}}+{\frac {6}{q^{25/2}}}-{\frac {10}{q^{23/2}}}+{\frac {13}{q^{21/2}}}-{\frac {14}{q^{19/2}}}+{\frac {14}{q^{17/2}}}-{\frac {12}{q^{15/2}}}+{\frac {8}{q^{13/2}}}-{\frac {6}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13}\left(-z^{3}\right)-3a^{13}z-a^{13}z^{-1}+3a^{11}z^{5}+12a^{11}z^{3}+12a^{11}z+2a^{11}z^{-1}-2a^{9}z^{7}-10a^{9}z^{5}-15a^{9}z^{3}-7a^{9}z-a^{7}z^{7}-5a^{7}z^{5}-8a^{7}z^{3}-5a^{7}z-a^{7}z^{-1}$ (db)
Kauffman polynomial	$a^{18}z^{4}-a^{18}z^{2}+3a^{17}z^{5}-3a^{17}z^{3}+a^{17}z+5a^{16}z^{6}-5a^{16}z^{4}+2a^{16}z^{2}+6a^{15}z^{7}-6a^{15}z^{5}+a^{15}z^{3}+a^{15}z+6a^{14}z^{8}-9a^{14}z^{6}+7a^{14}z^{4}-5a^{14}z^{2}+2a^{14}+4a^{13}z^{9}-4a^{13}z^{7}-2a^{13}z^{5}+3a^{13}z-a^{13}z^{-1}+a^{12}z^{10}+9a^{12}z^{8}-37a^{12}z^{6}+49a^{12}z^{4}-30a^{12}z^{2}+5a^{12}+7a^{11}z^{9}-22a^{11}z^{7}+25a^{11}z^{5}-22a^{11}z^{3}+13a^{11}z-2a^{11}z^{-1}+a^{10}z^{10}+5a^{10}z^{8}-30a^{10}z^{6}+41a^{10}z^{4}-21a^{10}z^{2}+3a^{10}+3a^{9}z^{9}-11a^{9}z^{7}+13a^{9}z^{5}-10a^{9}z^{3}+5a^{9}z+2a^{8}z^{8}-7a^{8}z^{6}+5a^{8}z^{4}+a^{8}z^{2}-a^{8}+a^{7}z^{7}-5a^{7}z^{5}+8a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}$ (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 -6                       1 1 -8                     2 1 -1 -10                   4     4 -12                 4 2     -2 -14               8 4       4 -16             6 4         -2 -18           8 8           0 -20         6 7             1 -22       4 7               -3 -24     2 6                 4 -26   1 4                   -3 -28   2                     2 -30 1                       -1

Integral Khovanov Homology (db, data source)

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

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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