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L11n406 L11n408

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X22,12,19,11 X10,4,11,3 X5,21,6,20 X21,5,22,18 X12,20,13,19 X14,9,15,10 X2,14,3,13 X8,15,9,16

Gauss Code:

{{1, -10, 5, -3}, {8, 6, -7, -4}, {-6, -1, 2, -11, 9, -5, 4, -8, 10, -9, 11, -2, 3, 7}}

Jones Polynomial:

- q-4 + 3q-3 - 5q-2 + 8q-1 - 10 + 11q - 9q2 + 9q3 - 5q4 + 3q5

A2 (sl(3)) Invariant:

- q-12 + q-10 + 2q-4 - 2q-2 + 1 + q4 + 5q6 + 2q8 + 6q10 + 3q12 + 3q14 + 4q16 + q18 + q20

HOMFLY-PT Polynomial:

a-6z-2 + a-6 - 2a-4z-2 - 4a-4 - 3a-4z2 - a-4z4 + a-2z-2 + 3a-2 + 3a-2z2 + 3a-2z4 + a-2z6 + 2z2 + 3z4 + z6 - 2a2z2 - a2z4

Kauffman Polynomial:

a-6z-2 - 4a-6 + 6a-6z2 - 2a-5z-1 + 4a-5z + 3a-5z5 + 2a-4z-2 - 6a-4 + 11a-4z2 - 12a-4z4 + 7a-4z6 - 2a-3z-1 + 2a-3z + 7a-3z3 - 16a-3z5 + 8a-3z7 + a-2z-2 - a-2 - 3a-2z2 + 4a-2z4 - 12a-2z6 + 6a-2z8 - 6a-1z + 20a-1z3 - 25a-1z5 + 4a-1z7 + 2a-1z9 + 3 - 14z2 + 33z4 - 32z6 + 9z8 - 6az + 18az3 - 10az5 - 3az7 + 2az9 + a2 - 6a2z2 + 17a2z4 - 13a2z6 + 3a2z8 - 2a3z + 5a3z3 - 4a3z5 + a3z7

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 11                   3 j = 9                 4 2 j = 7               5 1   j = 5             4 4     j = 3           7 5       j = 1         5 6         j = -1       3 5           j = -3     2 5             j = -5   1 3               j = -7   2                 j = -9 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 407]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 407]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[22, 12, 19, 11], > X[10, 4, 11, 3], X[5, 21, 6, 20], X[21, 5, 22, 18], X[12, 20, 13, 19], > X[14, 9, 15, 10], X[2, 14, 3, 13], X[8, 15, 9, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {8, 6, -7, -4}, > {-6, -1, 2, -11, 9, -5, 4, -8, 10, -9, 11, -2, 3, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-4 3 5 8 2 3 4 5 -10 - q + -- - -- + - + 11 q - 9 q + 9 q - 5 q + 3 q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 2 2 4 6 8 10 12 14 16 1 - q + q + -- - -- + q + 5 q + 2 q + 6 q + 3 q + 3 q + 4 q + 4 2 q q 18 20 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 407]][a, z]
Out[8]=	2 2 -6 4 3 1 2 1 2 3 z 3 z 2 2 4 a - -- + -- + ----- - ----- + ----- + 2 z - ---- + ---- - 2 a z + 3 z - 4 2 6 2 4 2 2 2 4 2 a a a z a z a z a a 4 4 6 z 3 z 2 4 6 z > -- + ---- - a z + z + -- 4 2 2 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 407]][a, z]
Out[9]=	4 6 -2 2 1 2 1 2 2 4 z 2 z 3 - -- - -- - a + a + ----- + ----- + ----- - ---- - ---- + --- + --- - 6 4 6 2 4 2 2 2 5 3 5 3 a a a z a z a z a z a z a a 2 2 2 3 6 z 3 2 6 z 11 z 3 z 2 2 7 z > --- - 6 a z - 2 a z - 14 z + ---- + ----- - ---- - 6 a z + ---- + a 6 4 2 3 a a a a 3 4 4 5 20 z 3 3 3 4 12 z 4 z 2 4 3 z > ----- + 18 a z + 5 a z + 33 z - ----- + ---- + 17 a z + ---- - a 4 2 5 a a a 5 5 6 6 16 z 25 z 5 3 5 6 7 z 12 z 2 6 > ----- - ----- - 10 a z - 4 a z - 32 z + ---- - ----- - 13 a z + 3 a 4 2 a a a 7 7 8 9 8 z 4 z 7 3 7 8 6 z 2 8 2 z 9 > ---- + ---- - 3 a z + a z + 9 z + ---- + 3 a z + ---- + 2 a z 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 3 2 5 3 5 5 q 6 q + 7 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 11 4 > 5 q t + 4 q t + 4 q t + 5 q t + q t + 4 q t + 2 q t + 3 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n407

L11n406 L11n408