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L11a139 L11a141

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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X22,10,7,9 X2738 X16,11,17,12 X12,5,13,6 X4,18,5,17 X14,19,15,20 X20,13,21,14 X18,22,19,21 X6,15,1,16

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 9q-11/2 - 16q-9/2 + 22q-7/2 - 26q-5/2 + 26q-3/2 - 24q-1/2 + 17q1/2 - 11q3/2 + 5q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 + 3q-20 - 2q-18 + 3q-14 - 5q-12 + 4q-10 + q-6 + 5q-4 - 3q-2 + 7 - 3q2 - q4 + 3q6 - 3q8 + q10

HOMFLY-PT Polynomial:

- a-1z3 - a-1z5 - az-1 + 4az3 + 3az5 + az7 + a3z-1 - 3a3z - 6a3z3 - 3a3z5 + 3a5z + 3a5z3 - a7z

Kauffman Polynomial:

- a-3z5 + 4a-2z4 - 5a-2z6 - 6a-1z3 + 15a-1z5 - 11a-1z7 + z2 - 4z4 + 16z6 - 13z8 + az-1 + az - 21az3 + 34az5 - 7az7 - 8az9 - a2 + 6a2z2 - 31a2z4 + 53a2z6 - 24a2z8 - 2a2z10 + a3z-1 + 5a3z - 29a3z3 + 29a3z5 + 9a3z7 - 14a3z9 + 8a4z2 - 34a4z4 + 47a4z6 - 18a4z8 - 2a4z10 + 6a5z - 21a5z3 + 20a5z5 + a5z7 - 6a5z9 + 2a6z2 - 9a6z4 + 14a6z6 - 7a6z8 + 2a7z - 7a7z3 + 9a7z5 - 4a7z7 - a8z2 + 2a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     4   j = 4                   7 1   j = 2                 10 4     j = 0               14 7       j = -2             13 11         j = -4           13 13           j = -6         10 14             j = -8       6 12               j = -10     3 10                 j = -12   1 6                   j = -14   3                     j = -16 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 140]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 140]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[2, 7, 3, 8], > X[16, 11, 17, 12], X[12, 5, 13, 6], X[4, 18, 5, 17], X[14, 19, 15, 20], > X[20, 13, 21, 14], X[18, 22, 19, 21], X[6, 15, 1, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -7, 6, -11}, > {4, -1, 3, -2, 5, -6, 9, -8, 11, -5, 7, -10, 8, -9, 10, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 9 16 22 26 26 24 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 17 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 11 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 3 2 3 5 4 -6 5 3 2 4 6 7 - q + --- - --- + --- - --- + --- + q + -- - -- - 3 q - q + 3 q - 20 18 14 12 10 4 2 q q q q q q q 8 10 > 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 140]][a, z]
Out[8]=	3 3 5 a a 3 5 7 z 3 3 3 5 3 z -(-) + -- - 3 a z + 3 a z - a z - -- + 4 a z - 6 a z + 3 a z - -- + z z a a 5 3 5 7 > 3 a z - 3 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 140]][a, z]
Out[9]=	3 2 a a 3 5 7 2 2 2 4 2 -a + - + -- + a z + 5 a z + 6 a z + 2 a z + z + 6 a z + 8 a z + z z 3 6 2 8 2 6 z 3 3 3 5 3 7 3 4 > 2 a z - a z - ---- - 21 a z - 29 a z - 21 a z - 7 a z - 4 z + a 4 5 5 4 z 2 4 4 4 6 4 8 4 z 15 z 5 > ---- - 31 a z - 34 a z - 9 a z + 2 a z - -- + ----- + 34 a z + 2 3 a a a 6 3 5 5 5 7 5 6 5 z 2 6 4 6 > 29 a z + 20 a z + 9 a z + 16 z - ---- + 53 a z + 47 a z + 2 a 7 6 6 8 6 11 z 7 3 7 5 7 7 7 8 > 14 a z - a z - ----- - 7 a z + 9 a z + a z - 4 a z - 13 z - a 2 8 4 8 6 8 9 3 9 5 9 2 10 > 24 a z - 18 a z - 7 a z - 8 a z - 14 a z - 6 a z - 2 a z - 4 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 3 1 6 3 10 6 12 14 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 10 14 13 13 13 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 7 t + 10 q t + 4 q t + 7 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a140

L11a139 L11a141