L11a29

L11a28

L11a30

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Link Presentations

[edit Notes on L11a29's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(5t(2)^{2}-8t(2)+5\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$22q^{9/2}-21q^{7/2}+14q^{5/2}-8q^{3/2}+q^{21/2}-4q^{19/2}+10q^{17/2}-15q^{15/2}+21q^{13/2}-24q^{11/2}+3{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}+a^{-9}z^{-1}-z^{5}a^{-7}+z^{3}a^{-7}+za^{-7}-a^{-7}z^{-1}-3z^{5}a^{-5}-6z^{3}a^{-5}-5za^{-5}-2a^{-5}z^{-1}-z^{5}a^{-3}+z^{3}a^{-3}+3za^{-3}+2a^{-3}z^{-1}+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-6}-2z^{10}a^{-8}-6z^{9}a^{-5}-13z^{9}a^{-7}-7z^{9}a^{-9}-8z^{8}a^{-4}-16z^{8}a^{-6}-16z^{8}a^{-8}-8z^{8}a^{-10}-6z^{7}a^{-3}-2z^{7}a^{-5}+17z^{7}a^{-7}+9z^{7}a^{-9}-4z^{7}a^{-11}-3z^{6}a^{-2}+11z^{6}a^{-4}+42z^{6}a^{-6}+48z^{6}a^{-8}+19z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}+9z^{5}a^{-3}+23z^{5}a^{-5}+8z^{5}a^{-7}+3z^{5}a^{-9}+8z^{5}a^{-11}+4z^{4}a^{-2}-7z^{4}a^{-4}-37z^{4}a^{-6}-44z^{4}a^{-8}-16z^{4}a^{-10}+2z^{4}a^{-12}+2z^{3}a^{-1}-9z^{3}a^{-3}-31z^{3}a^{-5}-19z^{3}a^{-7}-3z^{3}a^{-9}-4z^{3}a^{-11}-z^{2}a^{-2}-2z^{2}a^{-4}+14z^{2}a^{-6}+25z^{2}a^{-8}+9z^{2}a^{-10}-z^{2}a^{-12}-za^{-1}+7za^{-3}+15za^{-5}+9za^{-7}+2za^{-9}+2a^{-4}-4a^{-6}-9a^{-8}-4a^{-10}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-7}z^{-1}+a^{-9}z^{-1}$ (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 22                       1 -1 20                     3   3 18                   7 1   -6 16                 8 3     5 14               13 7       -6 12             11 8         3 10           11 13           2 8         10 11             -1 6       4 11               7 4     4 10                 -6 2   1 6                   5 0   2                     -2 -2 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ 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 r=-2} ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=6$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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