L11a193

L11a192

L11a194

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Link Presentations

[edit Notes on L11a193's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X16,6,17,5 X18,14,19,13 X20,16,21,15 X14,20,15,19 X12,22,13,21 X2738 X4,12,5,11 X6,18,1,17
Gauss code	{1, -9, 2, -10, 4, -11}, {9, -1, 3, -2, 10, -8, 5, -7, 6, -4, 11, -5, 7, -6, 8, -3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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