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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-11/2 - 3q-9/2 + 6q-7/2 - 11q-5/2 + 14q-3/2 - 17q-1/2 + 16q1/2 - 15q3/2 + 11q5/2 - 7q7/2 + 4q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-16 + q-14 - q-12 + 3q-8 - q-6 + 5q-4 + q-2 + 2 + 3q2 - 2q4 + 3q6 - 2q8 - q10 - 2q14 + q16

HOMFLY-PT Polynomial:

a-3z-1 + a-3z - 2a-3z3 - a-3z5 - 3a-1z-1 - 5a-1z + 3a-1z5 + a-1z7 + 2az-1 + 5az + 6az3 + 4az5 + az7 - 2a3z - 3a3z3 - a3z5

Kauffman Polynomial:

- 2a-5z3 + 3a-5z5 - a-5z7 + a-4 + 2a-4z2 - 15a-4z4 + 15a-4z6 - 4a-4z8 - a-3z-1 + a-3z + 3a-3z3 - 15a-3z5 + 17a-3z7 - 5a-3z9 + 3a-2 + 2a-2z2 - 25a-2z4 + 26a-2z6 - 3a-2z8 - 2a-2z10 - 3a-1z-1 + 7a-1z - a-1z3 - 18a-1z5 + 28a-1z7 - 10a-1z9 + 3 - 17z4 + 25z6 - 6z8 - 2z10 - 2az-1 + 8az - 15az3 + 12az5 + 3az7 - 5az9 - 2a2z2 - 2a2z4 + 9a2z6 - 7a2z8 + a3z - 6a3z3 + 9a3z5 - 7a3z7 - a4z2 + 4a4z4 - 5a4z6 - a5z + 3a5z3 - 3a5z5 + a6z2 - 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                     3   j = 8                   4 1   j = 6                 7 3     j = 4               8 4       j = 2             8 7         j = 0           9 8           j = -2         6 9             j = -4       5 8               j = -6     2 7                 j = -8   1 4                   j = -10   2                     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, 148]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 148]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[20, 11, 21, 12], > X[12, 6, 13, 5], X[4, 20, 5, 19], X[14, 18, 15, 17], X[16, 8, 17, 7], > X[22, 16, 7, 15], X[18, 14, 19, 13], X[6, 21, 1, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6, 5, -11}, > {8, -1, 2, -3, 4, -5, 10, -7, 9, -8, 7, -10, 6, -4, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 6 11 14 17 3/2 q - ---- + ---- - ---- + ---- - ------- + 16 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 - 7 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 -12 3 -6 5 -2 2 4 6 8 2 - q + q - q + -- - q + -- + q + 3 q - 2 q + 3 q - 2 q - 8 4 q q 10 14 16 > q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 148]][a, z]
Out[8]=	3 5 1 3 2 a z 5 z 3 2 z 3 3 3 z ---- - --- + --- + -- - --- + 5 a z - 2 a z - ---- + 6 a z - 3 a z - -- + 3 a z z 3 a 3 3 a z a a a 5 7 3 z 5 3 5 z 7 > ---- + 4 a z - a z + -- + a z a a
In[9]:=	Kauffman[Link[11, Alternating, 148]][a, z]
Out[9]=	2 -4 3 1 3 2 a z 7 z 3 5 2 z 3 + a + -- - ---- - --- - --- + -- + --- + 8 a z + a z - a z + ---- + 2 3 a z z 3 a 4 a a z a a 2 3 3 3 2 z 2 2 4 2 6 2 2 z 3 z z 3 3 3 > ---- - 2 a z - a z + a z - ---- + ---- - -- - 15 a z - 6 a z + 2 5 3 a a a a 4 4 5 5 3 4 15 z 25 z 2 4 4 4 6 4 3 z > 3 a z - 17 z - ----- - ----- - 2 a z + 4 a z - a z + ---- - 4 2 5 a a a 5 5 6 6 15 z 18 z 5 3 5 5 5 6 15 z 26 z > ----- - ----- + 12 a z + 9 a z - 3 a z + 25 z + ----- + ----- + 3 a 4 2 a a a 7 7 7 8 2 6 4 6 z 17 z 28 z 7 3 7 8 4 z > 9 a z - 5 a z - -- + ----- + ----- + 3 a z - 7 a z - 6 z - ---- - 5 3 a 4 a a a 8 9 9 10 3 z 2 8 5 z 10 z 9 10 2 z > ---- - 7 a z - ---- - ----- - 5 a z - 2 z - ----- 2 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 2 1 4 2 7 5 8 9 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 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 6 2 2 2 4 2 4 3 6 3 6 4 > ---- + 8 t + 8 q t + 7 q t + 8 q t + 4 q t + 7 q t + 3 q t + 2 q t 8 4 8 5 10 5 12 6 > 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a148

L11a147 L11a149