L10a94

L10a93

L10a95

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Link Presentations

[edit Notes on L10a94's Link Presentations]

Planar diagram presentation	X10,1,11,2 X14,5,15,6 X12,3,13,4 X4,13,5,14 X16,19,17,20 X18,7,19,8 X6,17,7,18 X20,15,9,16 X2,9,3,10 X8,11,1,12
Gauss code	{1, -9, 3, -4, 2, -7, 6, -10}, {9, -1, 10, -3, 4, -2, 8, -5, 7, -6, 5, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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