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
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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^{10} a^{-6} -z^{10} a^{-8} -2 z^9 a^{-5} -6 z^9 a^{-7} -4 z^9 a^{-9} -2 z^8 a^{-4} -4 z^8 a^{-6} -10 z^8 a^{-8} -8 z^8 a^{-10} -z^7 a^{-3} +2 z^7 a^{-5} +7 z^7 a^{-7} -6 z^7 a^{-9} -10 z^7 a^{-11} +7 z^6 a^{-4} +16 z^6 a^{-6} +22 z^6 a^{-8} +5 z^6 a^{-10} -8 z^6 a^{-12} +5 z^5 a^{-3} +12 z^5 a^{-5} +15 z^5 a^{-7} +24 z^5 a^{-9} +12 z^5 a^{-11} -4 z^5 a^{-13} -6 z^4 a^{-4} -8 z^4 a^{-6} -3 z^4 a^{-8} +8 z^4 a^{-10} +8 z^4 a^{-12} -z^4 a^{-14} -9 z^3 a^{-3} -23 z^3 a^{-5} -21 z^3 a^{-7} -14 z^3 a^{-9} -5 z^3 a^{-11} +2 z^3 a^{-13} -2 z^2 a^{-4} -6 z^2 a^{-6} -6 z^2 a^{-8} -5 z^2 a^{-10} -3 z^2 a^{-12} +7 z a^{-3} +13 z a^{-5} +8 z a^{-7} +3 z a^{-9} +z a^{-11} +3 a^{-4} +3 a^{-6} + a^{-8} -2 a^{-3} z^{-1} -3 a^{-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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