L10a114

L10a113

L10a115

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Link Presentations

[edit Notes on L10a114's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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