L11a130

L11a129

L11a131

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Link Presentations

[edit Notes on L11a130's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2(t(1)-1)(t(2)-1)\left(t(2)^{2}+1\right)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-12q^{9/2}+7q^{7/2}-5q^{5/2}+q^{3/2}+q^{23/2}-4q^{21/2}+8q^{19/2}-12q^{17/2}+15q^{15/2}-15q^{13/2}+15q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{5}a^{-9}+2z^{3}a^{-9}-a^{-9}z^{-1}-z^{7}a^{-7}-3z^{5}a^{-7}+6za^{-7}+5a^{-7}z^{-1}-z^{7}a^{-5}-5z^{5}a^{-5}-11z^{3}a^{-5}-14za^{-5}-8a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+8za^{-3}+4a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-14}+4z^{5}a^{-13}-2z^{3}a^{-13}+8z^{6}a^{-12}-8z^{4}a^{-12}+2z^{2}a^{-12}+10z^{7}a^{-11}-13z^{5}a^{-11}+4z^{3}a^{-11}+za^{-11}+8z^{8}a^{-10}-9z^{6}a^{-10}+3z^{2}a^{-10}-2a^{-10}+4z^{9}a^{-9}-5z^{5}a^{-9}-5z^{3}a^{-9}+3za^{-9}+a^{-9}z^{-1}+z^{10}a^{-8}+6z^{8}a^{-8}-10z^{6}a^{-8}-11z^{4}a^{-8}+20z^{2}a^{-8}-9a^{-8}+5z^{9}a^{-7}-10z^{7}a^{-7}+3z^{5}a^{-7}+3z^{3}a^{-7}-7za^{-7}+5a^{-7}z^{-1}+z^{10}a^{-6}-z^{8}a^{-6}+5z^{6}a^{-6}-25z^{4}a^{-6}+33z^{2}a^{-6}-14a^{-6}+z^{9}a^{-5}+z^{7}a^{-5}-15z^{5}a^{-5}+27z^{3}a^{-5}-21za^{-5}+8a^{-5}z^{-1}+z^{8}a^{-4}-2z^{6}a^{-4}-5z^{4}a^{-4}+14z^{2}a^{-4}-8a^{-4}+z^{7}a^{-3}-6z^{5}a^{-3}+13z^{3}a^{-3}-12za^{-3}+4a^{-3}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 24                       1 -1 22                     3   3 20                   5 1   -4 18                 7 3     4 16               8 5       -3 14             7 7         0 12           8 8           0 10         4 7             -3 8       3 8               5 6     2 4                 -2 4   1 5                   4 2                         0 0 1                       1

Integral Khovanov Homology (db, data source)

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

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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