L11a547

L11a546

L11a548

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Link L11a547.

A graph, L11a547.

A part of a knot and a part of a graph.

Link Presentations

[edit Notes on L11a547's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(4)^{2}t(3)^{2}+2t(2)t(4)^{2}t(3)^{2}-2t(4)^{2}t(3)^{2}+t(1)t(3)^{2}-t(1)t(2)t(3)^{2}+t(2)t(3)^{2}-2t(1)t(4)t(3)^{2}+2t(1)t(2)t(4)t(3)^{2}-3t(2)t(4)t(3)^{2}+3t(4)t(3)^{2}-t(3)^{2}-2t(1)t(4)^{2}t(3)+2t(1)t(2)t(4)^{2}t(3)-3t(2)t(4)^{2}t(3)+3t(4)^{2}t(3)-3t(1)t(3)+3t(1)t(2)t(3)-2t(2)t(3)+5t(1)t(4)t(3)-6t(1)t(2)t(4)t(3)+5t(2)t(4)t(3)-6t(4)t(3)+2t(3)+t(1)t(4)^{2}-t(1)t(2)t(4)^{2}+t(2)t(4)^{2}-t(4)^{2}+2t(1)-2t(1)t(2)+t(2)-3t(1)t(4)+3t(1)t(2)t(4)-2t(2)t(4)+2t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}+{\frac {4}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {4}{q^{25/2}}}+{\frac {9}{q^{23/2}}}-{\frac {17}{q^{21/2}}}+{\frac {21}{q^{19/2}}}-{\frac {27}{q^{17/2}}}+{\frac {25}{q^{15/2}}}-{\frac {24}{q^{13/2}}}+{\frac {16}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}+a^{13}z^{-3}+3z^{3}a^{11}+za^{11}-5a^{11}z^{-1}-3a^{11}z^{-3}-2z^{5}a^{9}+z^{3}a^{9}+11za^{9}+10a^{9}z^{-1}+3a^{9}z^{-3}-4z^{5}a^{7}-11z^{3}a^{7}-11za^{7}-5a^{7}z^{-1}-a^{7}z^{-3}-z^{5}a^{5}-z^{3}a^{5}$ (db)
Kauffman polynomial	$a^{16}z^{6}-2a^{16}z^{4}+a^{16}z^{2}+4a^{15}z^{7}-9a^{15}z^{5}+8a^{15}z^{3}-3a^{15}z+7a^{14}z^{8}-13a^{14}z^{6}+6a^{14}z^{4}+a^{14}z^{2}+6a^{13}z^{9}-23a^{13}z^{5}+25a^{13}z^{3}-a^{13}z^{-3}-12a^{13}z+5a^{13}z^{-1}+2a^{12}z^{10}+18a^{12}z^{8}-45a^{12}z^{6}+26a^{12}z^{4}+3a^{12}z^{2}+3a^{12}z^{-2}-10a^{12}+14a^{11}z^{9}-10a^{11}z^{7}-26a^{11}z^{5}+31a^{11}z^{3}-3a^{11}z^{-3}-21a^{11}z+12a^{11}z^{-1}+2a^{10}z^{10}+23a^{10}z^{8}-50a^{10}z^{6}+22a^{10}z^{4}+15a^{10}z^{2}+6a^{10}z^{-2}-19a^{10}+8a^{9}z^{9}+4a^{9}z^{7}-30a^{9}z^{5}+32a^{9}z^{3}-3a^{9}z^{-3}-23a^{9}z+12a^{9}z^{-1}+12a^{8}z^{8}-15a^{8}z^{6}+a^{8}z^{4}+12a^{8}z^{2}+3a^{8}z^{-2}-10a^{8}+10a^{7}z^{7}-17a^{7}z^{5}+17a^{7}z^{3}-a^{7}z^{-3}-11a^{7}z+5a^{7}z^{-1}+4a^{6}z^{6}-3a^{6}z^{4}+a^{5}z^{5}-a^{5}z^{3}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     4 1 -3 -8                   7     7 -10                 9 4     -5 -12               15 7       8 -14             12 11         -1 -16           15 13           2 -18         10 16             6 -20       7 11               -4 -22     3 11                 8 -24   1 6                   -5 -26   3                     3 -28 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-8$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=-7$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-6$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{15}$ $r=-5$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{15}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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