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L11a114 L11a116

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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X16,8,17,7 X22,18,5,17 X18,11,19,12 X20,9,21,10 X10,19,11,20 X8,21,9,22 X12,16,13,15 X2536 X4,13,1,14

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 2q-15/2 - 5q-13/2 + 8q-11/2 - 11q-9/2 + 14q-7/2 - 15q-5/2 + 13q-3/2 - 11q-1/2 + 7q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

q-28 + 2q-26 + q-22 + 2q-20 - 3q-18 - 3q-12 + 2q-10 + 3q-6 + 2q-4 - q-2 + 4 - 2q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 - az-1 - 2az - az3 - az5 + a3z-1 - a3z5 + a5z-1 + 3a5z + 3a5z3 - 2a7z-1 - 3a7z + a9z-1

Kauffman Polynomial:

- a-2z4 + 3a-1z3 - 4a-1z5 + 7z4 - 7z6 + az-1 - 3az + 9az5 - 8az7 - a2 + 4a2z2 - 6a2z4 + 10a2z6 - 7a2z8 + a3z-1 - 2a3z - 4a3z3 + 7a3z5 + 2a3z7 - 4a3z9 - 2a4 + 12a4z2 - 30a4z4 + 29a4z6 - 8a4z8 - a4z10 - a5z-1 + 8a5z - 11a5z3 - 4a5z5 + 15a5z7 - 6a5z9 - 3a6 + 12a6z2 - 25a6z4 + 20a6z6 - 3a6z8 - a6z10 - 2a7z-1 + 12a7z - 18a7z3 + 7a7z5 + 4a7z7 - 2a7z9 - a8 + 4a8z2 - 9a8z4 + 8a8z6 - 2a8z8 - a9z-1 + 5a9z - 8a9z3 + 5a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                       1 j = 4                     3   j = 2                   4 1   j = 0                 7 3     j = -2               7 5       j = -4             8 6         j = -6           6 7           j = -8         5 8             j = -10       4 7               j = -12     1 4                 j = -14   1 4                   j = -16   1                     j = -18 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, 115]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 115]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[16, 8, 17, 7], X[22, 18, 5, 17], > X[18, 11, 19, 12], X[20, 9, 21, 10], X[10, 19, 11, 20], X[8, 21, 9, 22], > X[12, 16, 13, 15], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -8, 6, -7, 5, -9, 11, -2, 9, -3, 4, -5, > 7, -6, 8, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 5 8 11 14 15 13 11 -q + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 7 Sqrt[q] - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 -22 2 3 3 2 3 2 -2 2 6 8 4 + q + --- + q + --- - --- - --- + --- + -- + -- - q - 2 q + 2 q - q 26 20 18 12 10 6 4 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 115]][a, z]
Out[8]=	3 5 7 9 3 a a a 2 a a 5 7 z 3 5 3 -(-) + -- + -- - ---- + -- - 2 a z + 3 a z - 3 a z + -- - a z + 3 a z - z z z z z a 5 3 5 > a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 115]][a, z]
Out[9]=	3 5 7 9 2 4 6 8 a a a 2 a a 3 5 -a - 2 a - 3 a - a + - + -- - -- - ---- - -- - 3 a z - 2 a z + 8 a z + z z z z z 3 7 9 2 2 4 2 6 2 8 2 3 z > 12 a z + 5 a z + 4 a z + 12 a z + 12 a z + 4 a z + ---- - a 4 3 3 5 3 7 3 9 3 4 z 2 4 4 4 > 4 a z - 11 a z - 18 a z - 8 a z + 7 z - -- - 6 a z - 30 a z - 2 a 5 6 4 8 4 4 z 5 3 5 5 5 7 5 > 25 a z - 9 a z - ---- + 9 a z + 7 a z - 4 a z + 7 a z + a 9 5 6 2 6 4 6 6 6 8 6 7 > 5 a z - 7 z + 10 a z + 29 a z + 20 a z + 8 a z - 8 a z + 3 7 5 7 7 7 9 7 2 8 4 8 6 8 > 2 a z + 15 a z + 4 a z - a z - 7 a z - 8 a z - 3 a z - 8 8 3 9 5 9 7 9 4 10 6 10 > 2 a z - 4 a z - 6 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 1 1 4 1 4 4 7 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 16 7 14 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 7 5 8 6 7 8 6 7 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 3 t + 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q t q t q t q t q t q t q t q t 2 2 2 4 2 6 3 > 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a115

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