L11a531

L11a530

L11a532

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Link Presentations

[edit Notes on L11a531's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{-2 u v w x^2+2 u v w x-u v w-u v x^3+3 u v x^2-3 u v x+u v-u w x^3+3 u w x^2-3 u w x+u w+2 u x^3-4 u x^2+4 u x-u-v w x^3+4 v w x^2-4 v w x+2 v w+v x^3-3 v x^2+3 v x-v+w x^3-3 w x^2+3 w x-w-x^3+2 x^2-2 x}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}}$ (db)
Jones polynomial	$-\frac{11}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{2}{q^{25/2}}+\frac{6}{q^{23/2}}-\frac{11}{q^{21/2}}+\frac{16}{q^{19/2}}-\frac{22}{q^{17/2}}+\frac{19}{q^{15/2}}-\frac{21}{q^{13/2}}+\frac{14}{q^{11/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{15} z^{-3} +5 a^{13} z^{-3} +5 a^{13} z^{-1} -9 a^{11} z^{-3} -10 a^{11} z-19 a^{11} z^{-1} +10 a^9 z^3+7 a^9 z^{-3} +26 a^9 z+23 a^9 z^{-1} -4 a^7 z^5-13 a^7 z^3-2 a^7 z^{-3} -16 a^7 z-9 a^7 z^{-1} -a^5 z^5-a^5 z^3$ (db)
Kauffman polynomial	$-z^6 a^{16}+4 z^4 a^{16}-6 z^2 a^{16}-a^{16} z^{-2} +4 a^{16}-2 z^7 a^{15}+5 z^5 a^{15}-4 z^3 a^{15}+2 z a^{15}-2 a^{15} z^{-1} +a^{15} z^{-3} -2 z^8 a^{14}-3 z^6 a^{14}+22 z^4 a^{14}-34 z^2 a^{14}-7 a^{14} z^{-2} +24 a^{14}-2 z^9 a^{13}-5 z^7 a^{13}+19 z^5 a^{13}-22 z^3 a^{13}+16 z a^{13}-12 a^{13} z^{-1} +5 a^{13} z^{-3} -z^{10} a^{12}-7 z^8 a^{12}+3 z^6 a^{12}+38 z^4 a^{12}-75 z^2 a^{12}-18 a^{12} z^{-2} +58 a^{12}-7 z^9 a^{11}-3 z^7 a^{11}+34 z^5 a^{11}-45 z^3 a^{11}+37 z a^{11}-24 a^{11} z^{-1} +9 a^{11} z^{-3} -z^{10} a^{10}-15 z^8 a^{10}+19 z^6 a^{10}+29 z^4 a^{10}-73 z^2 a^{10}-19 a^{10} z^{-2} +60 a^{10}-5 z^9 a^9-10 z^7 a^9+40 z^5 a^9-47 z^3 a^9+39 z a^9-23 a^9 z^{-1} +7 a^9 z^{-3} -10 z^8 a^8+10 z^6 a^8+12 z^4 a^8-26 z^2 a^8-7 a^8 z^{-2} +23 a^8-10 z^7 a^7+19 z^5 a^7-19 z^3 a^7+16 z a^7-9 a^7 z^{-1} +2 a^7 z^{-3} -4 z^6 a^6+3 z^4 a^6-z^5 a^5+z^3 a^5$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ 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                 7 4     -3 -12               14 7       7 -14             11 13         2 -16           11 8           3 -18         5 11             6 -20       6 11               -5 -22     1 6                 5 -24   1 5                   -4 -26   1                     1 -28 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-7$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-6$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-5$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-4$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{14}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

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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