L11a86

L11a85

L11a87

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Link Presentations

[edit Notes on L11a86's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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