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L11n112 L11n114

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X7,18,8,19 X19,22,20,5 X9,21,10,20 X21,9,22,8 X16,11,17,12 X14,17,15,18 X10,15,11,16 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 6q-11/2 - 9q-9/2 + 10q-7/2 - 11q-5/2 + 10q-3/2 - 8q-1/2 + 4q1/2 - 2q3/2

A2 (sl(3)) Invariant:

- q-24 - q-22 + q-20 - 2q-18 + 2q-16 + 3q-14 + 3q-10 - 2q-8 - q-4 - q-2 + 4 + 2q4 + 2q6

HOMFLY-PT Polynomial:

- 2a-1z-1 - 2a-1z + 5az-1 + 8az + 4az3 - 6a3z-1 - 11a3z - 7a3z3 - 2a3z5 + 4a5z-1 + 6a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial:

- 2a-1z-1 + 5a-1z - 3a-1z3 + 1 + z2 - 2z4 - z6 - 5az-1 + 20az - 26az3 + 12az5 - 4az7 + a2 - 10a2z4 + 9a2z6 - 4a2z8 - 6a3z-1 + 29a3z - 50a3z3 + 36a3z5 - 9a3z7 - a3z9 + 3a4 - 7a4z2 - 3a4z4 + 15a4z6 - 7a4z8 - 4a5z-1 + 18a5z - 35a5z3 + 33a5z5 - 8a5z7 - a5z9 + 3a6 - 9a6z2 + 8a6z4 + 4a6z6 - 3a6z8 - a7z-1 + 4a7z - 8a7z3 + 9a7z5 - 3a7z7 + a8 - 3a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 4                   2 j = 2                 2   j = 0               6 2   j = -2             6 4     j = -4           5 4       j = -6         5 6         j = -8       4 5           j = -10     2 5             j = -12   1 4               j = -14   2                 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, NonAlternating, 113]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 113]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[7, 18, 8, 19], X[19, 22, 20, 5], > X[9, 21, 10, 20], X[21, 9, 22, 8], X[16, 11, 17, 12], X[14, 17, 15, 18], > X[10, 15, 11, 16], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 6, -5, -9, 7, -2, 11, -8, 9, -7, 8, 3, > -4, 5, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 6 9 10 11 10 8 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 4 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 > 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 2 2 3 3 2 -4 -2 4 6 4 - q - q + q - --- + --- + --- + --- - -- - q - q + 2 q + 2 q 18 16 14 10 8 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 113]][a, z]
Out[8]=	3 5 7 -2 5 a 6 a 4 a a 2 z 3 5 7 3 --- + --- - ---- + ---- - -- - --- + 8 a z - 11 a z + 6 a z - a z + 4 a z - a z z z z z a 3 3 5 3 3 5 > 7 a z + 3 a z - 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 113]][a, z]
Out[9]=	3 5 7 2 4 6 8 2 5 a 6 a 4 a a 5 z 1 + a + 3 a + 3 a + a - --- - --- - ---- - ---- - -- + --- + 20 a z + a z z z z z a 3 3 5 7 2 4 2 6 2 8 2 3 z > 29 a z + 18 a z + 4 a z + z - 7 a z - 9 a z - 3 a z - ---- - a 3 3 3 5 3 7 3 4 2 4 4 4 > 26 a z - 50 a z - 35 a z - 8 a z - 2 z - 10 a z - 3 a z + 6 4 8 4 5 3 5 5 5 7 5 6 > 8 a z + 3 a z + 12 a z + 36 a z + 33 a z + 9 a z - z + 2 6 4 6 6 6 8 6 7 3 7 5 7 > 9 a z + 15 a z + 4 a z - a z - 4 a z - 9 a z - 8 a z - 7 7 2 8 4 8 6 8 3 9 5 9 > 3 a z - 4 a z - 7 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 2 1 4 2 5 4 5 6 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 5 6 5 4 6 2 4 2 > ----- + ----- + ----- + ---- + ---- + 2 t + 2 q t + 2 q t 6 3 6 2 4 2 4 2 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n113

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