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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X20,14,21,13 X6,19,1,20 X18,11,19,12 X12,6,13,5 X22,16,7,15 X4,18,5,17 X16,22,17,21

Gauss Code:

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

Jones Polynomial:

q-11/2 - 4q-9/2 + 8q-7/2 - 13q-5/2 + 16q-3/2 - 19q-1/2 + 17q1/2 - 15q3/2 + 11q5/2 - 6q7/2 + 3q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-16 + 2q-14 - q-12 + q-10 + 3q-8 - 2q-6 + 5q-4 + 3 + 2q2 - 3q4 + 2q6 - 3q8 + q12 - q14 + q16

HOMFLY-PT Polynomial:

- 2a-3z - 3a-3z3 - a-3z5 + 4a-1z + 6a-1z3 + 4a-1z5 + a-1z7 - az-1 - 3az + az3 + 3az5 + az7 + a3z-1 - 2a3z3 - a3z5

Kauffman Polynomial:

- 4a-5z3 + 4a-5z5 - a-5z7 + 6a-4z2 - 15a-4z4 + 12a-4z6 - 3a-4z8 - 3a-3z + 9a-3z3 - 16a-3z5 + 14a-3z7 - 4a-3z9 + 9a-2z2 - 22a-2z4 + 18a-2z6 - a-2z8 - 2a-2z10 - 5a-1z + 14a-1z3 - 25a-1z5 + 28a-1z7 - 10a-1z9 + 3z2 - 17z4 + 27z6 - 8z8 - 2z10 + az-1 - 2az - 8az3 + 14az5 + 2az7 - 6az9 - a2 - a2z2 - 2a2z4 + 13a2z6 - 10a2z8 + a3z-1 - 7a3z3 + 15a3z5 - 11a3z7 - a4z2 + 7a4z4 - 8a4z6 + 2a5z3 - 4a5z5 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 12                       1 j = 10                     2   j = 8                   4 1   j = 6                 7 2     j = 4               8 4       j = 2             9 7         j = 0           10 8           j = -2         7 10             j = -4       6 9               j = -6     3 8                 j = -8   1 5                   j = -10   3                     j = -12 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, 144]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 144]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 8, 15, 7], > X[20, 14, 21, 13], X[6, 19, 1, 20], X[18, 11, 19, 12], X[12, 6, 13, 5], > X[22, 16, 7, 15], X[4, 18, 5, 17], X[16, 22, 17, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -10, 8, -6}, > {4, -1, 2, -3, 7, -8, 5, -4, 9, -11, 10, -7, 6, -5, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 8 13 16 19 3/2 q - ---- + ---- - ---- + ---- - ------- + 17 Sqrt[q] - 15 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 11 q - 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 -12 -10 3 2 5 2 4 6 8 12 3 - q + --- - q + q + -- - -- + -- + 2 q - 3 q + 2 q - 3 q + q - 14 8 6 4 q q q q 14 16 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 144]][a, z]
Out[8]=	3 3 3 5 5 a a 2 z 4 z 3 z 6 z 3 3 3 z 4 z -(-) + -- - --- + --- - 3 a z - ---- + ---- + a z - 2 a z - -- + ---- + z z 3 a 3 a 3 a a a a 7 5 3 5 z 7 > 3 a z - a z + -- + a z a
In[9]:=	Kauffman[Link[11, Alternating, 144]][a, z]
Out[9]=	3 2 2 3 2 a a 3 z 5 z 2 6 z 9 z 2 2 4 2 4 z -a + - + -- - --- - --- - 2 a z + 3 z + ---- + ---- - a z - a z - ---- + z z 3 a 4 2 5 a a a a 3 3 4 4 9 z 14 z 3 3 3 5 3 4 15 z 22 z > ---- + ----- - 8 a z - 7 a z + 2 a z - 17 z - ----- - ----- - 3 a 4 2 a a a 5 5 5 2 4 4 4 6 4 4 z 16 z 25 z 5 3 5 > 2 a z + 7 a z - a z + ---- - ----- - ----- + 14 a z + 15 a z - 5 3 a a a 6 6 7 7 7 5 5 6 12 z 18 z 2 6 4 6 z 14 z 28 z > 4 a z + 27 z + ----- + ----- + 13 a z - 8 a z - -- + ----- + ----- + 4 2 5 3 a a a a a 8 8 9 9 7 3 7 8 3 z z 2 8 4 z 10 z 9 > 2 a z - 11 a z - 8 z - ---- - -- - 10 a z - ---- - ----- - 6 a z - 4 2 3 a a a a 10 10 2 z > 2 z - ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 3 1 5 3 8 6 9 10 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 7 2 2 2 4 2 4 3 6 3 6 4 > ---- + 8 t + 9 q t + 7 q t + 8 q t + 4 q t + 7 q t + 2 q t + 2 q t 8 4 8 5 10 5 12 6 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a144

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